In this paper (arXiv:1411.7569) we review two maximally superintegrable Hamiltonian systems that are defined, respectively, on an N-dimensional spherically symmetric generalization of the Darboux surface of type III and on an N-dimensional Taub-NUT space. Afterwards, we show that their quantization leads, respectively, to exactly solvable deformations of the two basic quantum mechanical systems: the harmonic oscillator and the Coulomb problem. In both cases the quantization is performed in such a way that the maximal superintegrability of the classical Hamiltonian is fully preserved. In particular, we prove that this strong condition is fulfilled by applying the so-called conformal Laplace-Beltrami quantization prescription. In this way, the eigenvalue problems for the quantum counterparts of these two Hamiltonians can be rigorously solved, and it is found that their discrete spectrum is just a smooth deformation of the oscillator and Coulomb spectrum, respectively. Moreover, it turns out that the maximal degeneracy of both systems is preserved under the deformation induced by the curvature. Finally, new multiparametric generalizations of both systems that preserve their superintegrability are envisaged.