Is quail and pigeon the same bird?

No-

They both come from the same family, and some (like white coturnix)look rather similar in body and beakshape, but they are completely different birds. Quail have different plumage, and cannot fly as well as pigeons, as they have very strong wing muscles built to take off quickly and fly over short distances, whereas pigeons can fly as well as most other columbiformes.