You can buy special low VOC paint or milk paint to be extra safe, but really most any ordinary acrylic/latex indoor paint will be just fine with your birds being on a different floor. I've painted the bird room and other rooms several times over the years with the birds often nearby. Even an open can of paint only has a mild odor and it dries very quickly (under an hour), after which there's no odor at all. Open some windows for ventilation until it's totally dry and you are good.
I'd probably avoid any oil based paints, though. They have much stronger fumes, take longer to dry and would be less safe.