The link between 3D spaces with (in general non-constant) curvature and quantum deformations is presented. It is shown, how the non-standard deformation of a sl(2) Poisson coalgebra generates a family of integrable Hamiltonians, that represent geodesic motions on 3D manifolds with a non-constant curvature, that turns out to be a function of the deformation parameter z. A different Hamiltonian defined on the same deformed coalgebra is also shown to generate a maximally superintegrable geodesic motion on 3D Riemannian and (2+1)D relativistic spaces whose sectional curvatures are all constant and equal to z. This approach can be generalized to arbitrary dimension.