We generate correlated photon pairs in an all-pass silicon microring resonator using continuous-wave pumped spontaneous four-wave mixing and we characterize in detail the optical bistability in the nonclassical single-photon regime. Special attention is given to the resonance-locked method by balancing the pump-induced heating and the active off-chip cooling, leading to stable operation of the photon-pair source. The maximal coincidence rate of 20.53 ± 0.34 Hz, maximal coincidence-to-accidental ratio of 654 ± 125, and minimal zero-delayed heralded second-order correlation of 0.14 ± 0.09 are achieved. The wavelengths of photon pairs remain unchanged at different pump power, indicating that our source is more compatible with the common communication systems than those using conventional on-resonance strategies, that tunes the pump wavelength. This work not only experimentally validates the nonclassical bistability behaviors, but also puts forward a simple method of pump-resonance matching for high-quality performance.