We propose a scheme for quantum-light generation in a nonlinear cavity hybridized with a two-level system and theoretically show that, when excited by a series of controlled pump pulses, the hybrid source generates Fock states with high probabilities, e.g., one- and two-photon states can be generated near on demand, and Fock states with up to seven photons with a probability above 50%. The tailorable nature of the nonlinear cavity allows for generating Fock states with arbitrary frequencies, even with a fixed two-level system, creating fundamentally new opportunities in all areas of quantum technologies.