By extending the quantum mechanical calculation for a proton motion on the one-dimensional O-H-O bond to the three-dimensional bond network, the possible proton hopping paths and the diffusion constants are explained in the acceptor-doped perovskite-oxides (ex. Y-doped SrZrO3). In the perovskite structure with an octahedral O-O network, the stable position of the proton is calculated to compare with neutron scattering data, and combined to make the simple diffusion path by taking into account the activation energy for each process. The numerical results on proton diffusivity are compared with some kinds of experimental data in good agreement. Based on this theoretical analysis, it seems that the proton diffusion mechanism is made clear through a proton hopping motion on the temporary-made O-H-O bond in the corner-linked network of the O-octahedron. Furthermore, by extending the proton diffusion to Li+ diffusion in perovskite-like oxides (ex. alpha-MoO3) with edge-sharing network of O-octahedron, T-dependence of the ion diffusivity is discussed on the analogy between H+ and Li+ diffusion mechanism.