  Given two c-projectively equivalent metrics on a K\"ahler manifold we show
that the canoncially constructed, Poisson-commuting integrals of motion of the
geodesic flow, linear and quadratic in momenta, also commute as quantum
operators. The methods employed here also provide a proof of a similar
statement in the case of projective equivalence. We also investigate the
addition of potentials, i.e. the generalization to natural Hamiltonian systems.
We show that the commuting operators lead to separation of variables for
Schr\"odinger's equation.
