If by conflict you are referring to the traditional "x versus y" then there are several conflicts going on with Emily. The "versus" is the opposing force.
Emily vs. her father- Old man Grierson, who raised Emily basically in isolation from the rest of society during his lifetime, embedded into Emily's subconscious that nobody was good enough for her. He shut down any chance for Emily to connect with a man (or anyone for that matter) in a healthy way. Moreover, he was brash and antisocial, creating the same conflicts in his daughter. The result was that Emily became an eccentric and lonely misfit that later on became the talk of the town. This is far from the dignified status one hopes to achieve at an older age. Moreover, this deprivation causes Emily to foster a bad tendency to react to men. She can neither keep them nor let them go. All of these things were caused directly by her father's extreme over-protection.
Emily vs. Homer- Homer Barron, the Yankee and drifter who comes during the re-construction of Jefferson County, is a man who has the same social faults as Emily, but is in no way a marrying man, nor someone who could fit the place of a beau for this lonely woman. He would naturally want to leave town once his business is done there, leaving Emily alone after a considerable amount of time "dating". This causes Emily to take matters into her own hands. Not only does she buy him an entire marriage trousseau as if he were to become a groom, but she lures him back into town only for him to never be seen alive again...that is, until his carcass is found in Emily's bedroom decades later.
Emily vs. Jefferson County- The fact that nothing has changed in Emily's life, her home, her mind, or anything even though times have moved on in the county, tells a lot about the state of mind of the woman. She refuses to pay taxes in the place in lieu of a promise of honor that the late Colonel Sartoris made on behalf of Emily's father, which exempted old man Grierson from paying taxes due to "favors" done for the town. This, however, was years ago.
While everyone involved in this past deal is now dead, Emily carries on as if time had stood still. Hence, she comes in direct conflict with her setting, which is desperately trying to move on, and away from the past, while she does the exact opposite.
Emily vs. her relatives- Emily's cousins had been summoned from Alabama to try and figure out what was the deal between Emily and Homer, especially when the later was making a spectacle of herself with the man. It is apparent that they did very little to change things. Moreover, they were not likable according to the townsfolk narrator, who tells us how glad everyone was when they were finally gone.
We were really glad. We were glad because the two female cousins were even more Grierson than Miss Emily had ever been.
Emily vs. herself- It is clear that there is a lot of internal conflict that stems from Emily's upbringing and continues onto her adulthood. In fact, this internal conflict is what ultimately makes Emily become her worst enemy.
The woman was not happy. From the way that she becomes bound to Homer we can see how desperately she wants to form normal and healthy relationships. She screams for help without saying anything.
First, she breaks down completely when her father dies and refuses to give up his body. Then, she gets very ill and resurfaces changed, but only externally. She attempts to make connections by offering pottery painting classes, but shortly shuts down and ends that up as well. Emily HAS tried to come into normalcy, but she has not succeeded. Sadly, the only thing she can do is take matters into her own hands and change some factors around her, rather than changing her own frame of mind. This is indicative of a personal struggle that she is too angry, too frustrated, or too scared to face.