The unwanted visitors have become an irritation to people in Low Catton, near Stamford Bridge, since arriving in July.
As well as their loud screeching, resident David Gilbertson said the two male birds are sending his dog "bananas," adding: "They have spent more time on my garden furniture than I have this summer."
The RSPCA won't deal with the birds, and East Riding Council said moving them wasn't in their remit.
Mr Gilbertson said: "They seem to have become very attached to our locality. They make a mess and I am sure there is somewhere better for them.
"Some people like having them round but most don't - they are scared of them or are concerned about the potential damage and mess they make.
He rang the RSPCA, but they "didn't want to know" as the birds were uninjured.
Councillor Andy Strangeway was called in to help. He said: "As numerous villagers do not like the peacocks I have requested East Riding Council to relocate them to somewhere where they will be more appreciated."
But a council spokesman said they had yet to receive any complaints about the birds.
He added: "However, as the birds are likely to be undomesticated, it is not within the council's remit to relocate them."