The conflicts can only be between Montresor and Fortunato, since they are the only two characters in the story. A conflict in a story arises when one character, the protagonist, wants something and is having a hard time getting it, for one reason or another. Montresor wants to murder Fortunato, but he wants to do so with what he calls "impunity." He not only doesn't want to get caught and punished for the crime, but he doesn't even want to be suspected. For years he puts up with Fortunato's "injuries" and "insults" and pretends to be the man's friend. He refers to Fortunato many times as "my friend" and even once or twice as "my good friend." This is because he has formed the habit of doing so publicly as often as possible, and now he keeps doing it out of force of habit. When Fortunato disappears, no one will suspect Montresor of foul play because the two men were known to be such good friends. This is one way Montresor solves a problem or conflict.
He must steer Fortunato down into his underground vaults. How? This is another conflict. He invents a cask of amontillado to lure his victim to his palazzo. This works--but he doesn't want to be seen on the streets with Fortunato shortly before the man's disappearance. How can he avoid being noticed on the crowded streets during the riotous carnival season? This is another conflict. Poe resolves this by dressing Fortunato in the most conspicuous possible costume, as a jester who even has a cap ringing bells on his head. Naturally this would attract attention, but it would distract attention from the companion who was wearing a black cloak and a black mask.
Then Montresor has to guide Fortunato into a narrow niche and attach the chains before his victim understands what is happening and puts up a struggle. Montresor handles this conflict by keeping Fortunato drunk. Twice they stop and consume bottles of wine. No doubt Montresor makes sure his "good friend" gets the most of each bottle.
After Fortunato is chained to the wall, the major conflicts are solved. The victim screams for help, but nobody can hear him so deep underground. He tries to talk Montresor into releasing him, but Montresor is adamant. He pleads for mercy, but these pleas are only music to Montresor's ears.
No one ever suspects Montresor of being responsible for Fortunato's mysterious disappearance, and the body has remained undiscovered for fifty years. Montresor had many conflicts to resolve, and he did so successfully with planning and foresight.