Welcome to BYC! Please make yourself at home and we are here to help.
I always think that keeping birds with dogs or cats is a bad idea. Cats and dogs usually see birds as dinner. So even if your cat seems ok with the quail you need to take extra caution. If you are lucky, the cat and quail will get along just fine. Just be careful.
Depends on how well you can keep your cat away from your bird. It will always be looking for a chance to nail him. If the bird cage isn't properly closed, or the little type slips through the bars - he's toast.
Welcome to BYC. Glad you decided to join our flock. You can only keep the quail with the cat if the quail is secured in such a way that it is impossible for the cat to physically make contact with the quail. If you keep your quail in a flight cage, you need to be certain that the cage is either completely out of reach of the cat or else that it is impossible for the cat to get a paw through the bars of the cage or get the cage door open. Remember that cats are hunters by nature, and given any opportunity, your cat will kill your quail. Please feel free to ask any other questions you may have. We are here to help in any way we can. Good luck with your quail.
You should be able to keep Button quail and your cat as long as you never let the two meet. Button quail do not like to be handled, so you won't be able to bring them out of the cage much. If you do, make SURE to keep the cat locked away. Always keep the cage somewhere safe in case the cat jumps on the cage and knocks it over.
Good luck in what ever you decide and welcome to our flock!