I'm not sure you can pinpoint a single conflict in "A Rose for Emily." Emily's refusal to adhere to the changing world around her is a general problem that seems to affect her throughout her life. She considers herself a little better than everyone else, and the townspeople believed that
... the Griersons held themselves a little to high for what they really were.
This attitude is evident in the major conflicts of the story: She refuses to pay her taxes because her father had been exempted many years before. Her last-ditch desire to marry leads her to courting the Yankee commoner, Homer Barron--a man of whom her father would never approve. She flaunts their courtship in the face of the disapproving townspeople; and she kills Homer after he refuses to marry her, keeping the body close to her--a perverse decision by any standard.