O'Connor was a hard-core bird lover. Her favorite bird? Yes, the peacock. She owned dozens of them and considered herself their servant.
In her November 25, 1955 letter to her friend Betty Hester, O'Connor discussed the peacock in "The Displaced Person":
The Priest sees the peacock as standing for the Transfiguration [one of Christ's stages of transformation], for which it is most certainly a most beautiful symbol. It also stands in medieval symbology for the Church – the eyes are the eyes of the Church. (source: Collected Works, 971)
Notice that O'Connor doesn't try to tell us how to read the peacock in the story, but rather to share with us what it means to her. The quote suggests that she uses the peacock to let us know what she thinks of a character. For example Father Flynn and Astor are the only characters who seem to care about the peacocks: the priest is fixated on the bird, and Astor is upset about the fact that Mrs. McIntyre starved the peacock population from twenty plus to one peacock and two peahens. That makes Mrs. McIntyre a villain and Father Flynn and Astor likeable characters. Notice also that neither Father Flynn nor Astor are present at the murder of Mr. Guizac.
Most of the characters in "The Displaced Person" don't seem to notice the peacock at all. It seems that O'Connor uses the peacock when she wants to make a specific point about a certain character or set of characters. The other character strongly associated with the peacock is Mrs. Shortley. Her relationship with it is ambiguous. She isn't fascinated by it and doesn't appreciate it. The thing that makes the relationship ambiguous is the fact that the peacock follows her around and seems to be fascinated by her. The "fiery wheels with fierce dark eyes in them" sound suspiciously like the eyes on the peacock's tail. If these are the eyes of the Church then does that mean the Church has its eyes on Mrs. Shortley? If so, what does that mean? It depends, we suppose, on how the Church is looking at her. Disapprovingly? Lovingly? We leave it to you to decide.