What is a major conflict between Emily and her father?
It is also possible that Emily and her father were engaged in an incestuous relationship. He chased away all of Emily's suitors, keeping her to himself. In the portrait of Emily and her father, Emily is wearing a white dress, indicating her possible role as bride/wife to her father. Faulkner also may have been implying incest among the Griersons with the mention of "insanity in the family" (sometimes a consequence of incestuous marriages/offspring). Finally, we know that Emily was in denial about her father's death and that she kept her father's body longer than she should have. Emily also keeps (her lover) Homer's body, suggesting a parallel between the two. Playing roles of father/daughter/lover would definitely result in conflict between Emily and her father.
The reader is never a witness to any interaction between Miss Emily and her father. But the narration gives us some indication of what possible conflict could have existed between the father and daughter.
Emily's father chased away every possible suitor that came to call on Emily because he felt that none of them were good enough. It can be understood that this must have irked a young girl to have this done. Other girls her age were out and dating, getting married, and having families. The world passed Emily by, and there was most likely some resentment.
It can also be seen that her relationship with Homer was a form of rebellion. Although her father was dead, she would have known how much he would have hated her seeing him.