Faulkner's "A Rose for Emily" presents conflict between the old South and the new South, between the status quo and change, between Emily and the townspeople around her.
I don't know if there is one specific point you can cite and say that that is the main place at which the conflict is stated. The conflict builds as incidents and situations are revealed. The conflict isn't a simple human vs. human conflict, for instance. No one main argument or fight demonstrates the entire conflict.
You could point to specific incidents, though, any of which demonstrate the conflict:
- Townspeople trying to get Emily to pay real estate taxes.
- Townspeople trying to update the house so mail carriers can deliver mail to it.
- Townspeople trying to get rid of the nasty smell coming from Emily's house.
- Emily refusing to give up her father's body.
- The discovery of Homer's body and the indentation on the pillow and the hair.
Emily represents the old South before the Civil War. She refuses to change and live as anything other than a plantation owner. All of the above indicate her refusal to adjust to the times and her situation, and indicate her conflict with those around her.