done the wrong.

In more simple language, he wants to kill Fortunato without getting caught. The word "impunity" suggests that he wants to feel completely safe not only from exposure but even from the slightest suspicion of being guilty of the crime. He also wants his victim to be sure that it is he, Montresor, who is responsible for his victim's murder. In other words, he doesn't want to do something like sending Fortunato a bottle of poisoned wine or hiring some assassin to kill Fortunato in a dark alley. Montresor wants to do the deed himself with impunity and have Fortunato--but only Fortunato--aware that he is killing him for revenge.

Montresor has a lot of problems achieving his perfect revenge, and his coping with all his problems is the essence of the story. He lures Fortunato down into the catacombs and chains him to the granite wall. He manages to do this without being recognized by anyone in the streets above. Fortunato was drunk when Montresor encountered him, and Montresor keeps him drunk until he has him in chains. Then Fortunato realizes he is in deep trouble and quickly sobers up. This was necessary for Montresor to be sure that Fortunato knows what is happening, why it is happening, and who is doing it. 

It is noteworthy that Fortunato never calls Montresor by name until he is chained to the granite wall. Then he cries:

"For the love of God, Montresor!”

“Yes,” I said, “for the love of God!”

Montresor says "Yes," because he now has proof that Fortunato recognizes him as his killer. Montresor adds, "...for the love of God!" because he is pleased to have his arrogant victim begging for mercy.

At the very end of the tale Montresor states:

Against the new masonry I re-erected the old rampart of bones. For the half of a century no mortal has disturbed them. In pace requiescat!

The fact that no one has discovered the bones in fifty years proves that no one has ever discovered the wall or ever suspected what is behind it. Montresor has achieved his revenge with "impunity." The purpose of getting revenge was to rid himself of all the painful thoughts and feelings that made him want the revenge in the first place. Now that he is fully satisfied, he means it sincerely when he says, "In pace requiescat!" (Rest in peace.) The Latin words are meant to convey the idea to the reader that Montresor did achieve the kind of revenge he wanted and feels utterly cleansed of his hatred and rage.

In Poe's classic short story "The Cask of Amontillado," Montresor does not go into specific detail regarding the exact reason he decides to murder Fortunato by entombing him alive inside his family's vast catacombs. In the first paragraph of the story, Montresor mentions that Fortunato had caused him a "thousand injuries" and "ventured upon insult," which motivated him to take revenge. Although Montresor is being vague and ambiguous, the reader can infer that he is a proud man who is extremely sensitive and vengeful. As the story progresses, Montresor approaches Fortunato, who casually insults Luchesi without hesitation. Fortunato's arrogant demeanor and flippant insult suggest that he may have done the same to Montresor. If Fortunato is willing to slander Luchesi behind his back, it is likely that he did the same to Montresor.

There is also evidence to suggest that Fortunato may have undermined Montresor's business ventures. The reader is aware that Fortunato is a rich man, and a pipe is a rather large quantity of wine. When Montresor informs Fortunato that he has purchased a pipe of the extremely rare Amontillado, Fortunato jumps at his chance to authenticate the wine in hopes of purchasing more. Fortunato recognizes that Amontillado is extremely rare during the carnival season and hopes to take advantage of the situation. Fortunato can certainly afford the great quantity of wine and make a profit selling the Amontillado.

If Fortunato plans on authenticating the wine to purchase more of the Amontillado before the merchant ship departs, the theory that he previously undermined Montresor's business practices holds weight. Additionally, Montresor is no longer the wealthy man he once was and may blame Fortunato for his financial struggles. However, we can never be certain of Montresor's exact motivation based on such circumstantial evidence.

Montresor is very serious about the subject of revenge and views it as a moral obligation to right the wrongs suffered by one's enemy. Montresor's ideology concerning revenge corresponds to the motto written on his family's coat of arms, which reads, "Nemo me impune lacessit," or, "No one provokes me with impunity." Montresor also has a precise definition of revenge, which he describes in the first paragraph of the short story. Montresor mentions that, when exacting revenge, it is imperative that one "must not only punish but punish with impunity." He is essentially saying that it is necessary not only to punish one's enemy but also to do so without being caught or suffering the consequences of exacting revenge. Montresor goes on to say that an avenger will not feel satisfied unless the victim knows who is responsible for their punishment. After describing the ideal conditions involved in avenging one’s enemy, Montresor elaborates on how he committed the perfect crime by getting revenge on Fortunato. Montresor's calculating scheme and horrific crime emphasize the seriousness of his attitude toward revenge. Montresor's actions correspond to his definition perfectly, and Fortunato's fate is disturbing and shocking.

"The Cask of Amontillado" is all about Montresor's revenge: the terrible revenge he wreaks upon the hapless, misnamed Fortunato. We're never told exactly what Fortunato has done to Montresor that warranted being walled up alive inside the catacombs. But whatever it was, this unspecified insult has driven Montresor into a frenzy that can only be satisfied by some serious payback—the most terrible revenge you could possibly imagine.

In his warped imagination, Montresor feels that revenge is not just necessary, it is the right thing to do. Revenge is like a moral imperative for Montresor, something he simply has to do if he is to redress the "thousand injuries" that Fortunato has heaped upon him over the years. But if Monstresor's going to gain revenge, he needs to do it properly. That means making sure that he gets away with his crime. His elaborate revenge plot has clearly been carefully planned and worked out to the last detail. This is to ensure no one will ever suspect Montresor of having carried out this wicked, dastardly deed.

In "The Cask of Amontillado," Montresor never gives a specific reason as to why he wants revenge. Rather, all that he tells us is that he had some personal slight against Fortunato. He claims that Fortunato had wronged him, grievously in his own mind.

Furthermore, if we are to believe his own words, it is certainly implied that this grievance had grown over time. Consider the very first sentence which opens the story: Montresor tells us, "the thousand injuries of Fortunato I had borne as best I could, but when he ventured upon insult, I vowed revenge." This unstated "insult" which he claims Fortunato had visited upon him, whatever it might have been, seems like it was more than anything else, a final straw, pushing him past his breaking point. Whatever ill will he held against Fortunato, it held much deeper roots reaching back in time.

That's all we can really say with any certainty.

Poe doesn't specify exactly why Montresor wants revenge. All that we are told is that Fortunato insulted Montresor in some unspecified manner and has subjected his nemesis to a thousand injuries. In examining Montresor's motives we need to bear in mind that we only hear his side of the story. There could be all sorts of reasons why he would choose to exact such a terrible revenge: some trivial, some more serious. At the very least, there's no doubt that Montresor feels deeply hurt and offended by Fortunato and is itching to have his revenge. That he should choose to exact this revenge in such a grotesquely horrific fashion indicates that, for Montresor at least, this is a very serious matter indeed.

It is that he has no bona fide cause for revenge (one that he states) that makes this story so horrifying.  By not identifying the cause for revenge, we cannot dismiss it as unworthy or outside of our range of experience.  Instead, the revenge achieves a universality, a flaw of human character, or perhaps more aptly, a commonality of human emotion that Montressor works out for us that we, only in our unconscious, would dare to think about.  Poe often treats such taboo subjects that he considers fundamental to human experience but too awful for a person to lay claim to.  Indeed, in an essay on poetry he calls such horrific experiences and emotions "beautiful."

Unfortunately, the answer to your question is no. Montresor does not detail his reason for revenge other than to say that Fortunato had "done him a thousand injuries". He doesn't detail any single one, nor does he ever give any specific reason for his cruel revenge. It could be assumed that, as in many of Poe's tales, there is not actually any wrong deed done to lead to the horrible action, but there is nothing in writing to prove that. 

Montresor believes that a person should take revenge in a calculated way.

Montresor states:

A wrong is unredressed when retribution overtakes its redresser. It is equally unredressed when the avenger fails to make himself felt as such to him who has done the wrong. (enotes e-text p. 4)

In order to get “redress” (translation: payback), one needs to take time and make sure that the person who is in the wrong knows that he is taking revenge.  Montresor plans to punish Fortunado "with impunity" (in other words, without risk of punishment).  Montressor identifies Fortunado’s weak point (wine) and uses that to get him.

Monresor claims that he waited patiently for revenge.  Montesor believes that revenge best when the victim does not suspect it.  He is willing to wait, to take his time, to make sure his victim is off guard.  As he points out, he does not make threats.  He waits.

You, who so well know the nature of my soul, will not suppose, however, that I gave utterance to a threat. At length I would be avenged; this was a point definitely, settled—but the very definitiveness with which it was resolved precluded the idea of risk. (enotes e-text p. 4)

Basically, he is saying that he took the time to make a good plan, and therefore there was no risk in carrying it out.