Thanks for your question.
There are very few diseases that will be transmitted from cats to birds other than bacteria transmitted through bite wounds. Cats are predators and should never be trusted alone with birds, even those they seem friendly with. Bite wounds essentially inject bacteria from the cat’s mouth into the bird and the infection is frequently fatal, even if the initial bite trauma is mild.
Quakers are great pets. Be sure to have your new pet fully examined by an experienced avian veterinarian after you obtain it!