There is certainly plenty of conflict in this excellent short story, both internal and external. However, the conflict that I think your question alludes to is the internal conflict that goes on in Miss Emily herself, a character that we only get to see through the eyes of other people and never from her own perspective. We are but given tantalising glimpses of her background, that mostly focuses on gossip and hearsay, however, one distinct memory that we are given is the strict and authoritarian way in which her father raised her, scaring away any suitors with his whip:
We had long thought of them as a tableau, Miss Emily a slender figure in white in the background, her father a spraddled silhouette in teh forground, his back to her and clutching a horsewhip, the two of them framed by the back-flung front door.
It is clear from this "tableau" that Miss Emily had a very difficult childhood in a sense, and was kept isolated from forming human attachments with others thanks to her father. However, as the story progresses, we continue to see how Miss Emily becomes ever more stranded from the rest of humanity, and how, in a sense, she is already living a life that is dead, as her description as a "bloated" corpse, "like a body long submerged in motionless water," indicates. As we discover the grisly murder that she committed we realise the conflict that Miss Emily has undergone. Having had love denied to her for so long, she finally receives it, only to face losing it. Her desire for companionship of any kind, whatever the price, is evident in the strands of grey hair that adorn the pillow next to the corpse of Homer Barron. Through killing him, she gains him, but also marks her own exit from the world through her corpse-like appearance. Love is so strong sometimes that it expresses itself in acts of violence, and even of murder.