We demonstrate the generation of quantum-correlated photon pairs combined with the spectral filtering of the pump field by more than 95 dB on a single silicon chip using electrically tunable ring resonators and passive Bragg reflectors. Moreover, we perform the demultiplexing and routing of signal and idler photons after transferring them via an optical fiber to a second identical chip. Nonclassical two-photon temporal correlations with a coincidence-to-accidental ratio of 50 are measured without further off-chip filtering. Our system, fabricated with high yield and reproducibility in a CMOS-compatible process, paves the way toward large-scale quantum photonic circuits by allowing sources and detectors of single photons to be integrated on the same chip.