In perovskite-type oxide based protonic conductors (ex. Sc-doped SrTiO3), a quantum mechanical calculation is developed to suggest the conduction mechanism. By extending one-dimensional O-H(D)... O hydrogen bonds to a three-dimensional bond network, the zero-point vibration and tunneling motion of a proton (or deuteron) on the bond are investigated in the perovskite structure. Numerical results are given and used to interpret the H(D)-isotope effect on the electrical conductivity and the infrared absorption spectra.