It is possible to determine some eigenvalue of momentum for an electron in an atom, but you can't take that and do celestial mechanics with it, because the sharpness in knowing the velocity comes at the cost of vagueness in knowing the position. You can't have a classical orbit without knowing both, and the uncertainty principle says you can't know both accurately at the same time.
Because of uncertainty and its pal the exclusion principle, electrons in atoms arrange themselves in shells, the number of shells increasing through the rows of the periodic table. The outer shells are at a higher energy level than the inner ones, but again there are no Kepler's laws to resort to in quantum land.
Though it´s not appropriate to say ¨the velocity¨ of the electrons in an atom, there´s still semi-classical analog: probability flow density. Also, you can mimic the conception of velocity as p/m, where p is the momentum operator, take the expectation value of p/m, you can get a semi-classical velocity.