We theoretically propose quantum spin pumping mediated by magnons, under a time-dependent transverse magnetic field, at the interface between a ferromagnetic insulator and a non-magnetic metal. The generation of a spin current under a thermal equilibrium condition is discussed by calculating the spin relaxation torque which breaks the spin conservation law for conduction electrons, and the torque operates the coherent magnon state. Localized spins lose spin angular momentum by emitting magnons and conduction electrons flip from down to up by absorbing the momentum. The spin relaxation torque has a resonance structure as a function of the angular frequency of the applied transverse field. This fact is useful to enhance the spin pumping effect induced by quantum fluctuations. We also discuss the distinction between our quantum spin pumping theory and the one proposed by Tserkovnyak et al.