We present a quantum-mechanical theory to describe narrow-band photon-pair generation via four-wave mixing in a Silicon-on-Insulator (SOI) micro-resonator. We also provide design principles for efficient photon-pair generation in an SOI micro-resonator through extensive numerical simulations. Microring cavities are shown to have a much wider dispersion-compensated frequency range than straight cavities. A microring with an inner radius of 8 mu m can output an entangled photon comb of 21 pairwise-correlated peaks (42 comb lines) spanning from 1.3 mu m to 1.8 mu m. Such on-chip quantum photonic devices offer a path toward future integrated quantum photonics and quantum integrated circuits.