We study quantum stochastic resonance (QSR) in the periodically driven biased spin-boson system. The amplitude and the phase of the nonlinear response function vs temperature are studied in the incoherent and coherent tunneling regimes, encompassing adiabatic and nonadiabatic driving, weak and strong Ohmic coupling α. For α = 1/2 we present exact QSR results. As a general feature, a principal maximum appears when the static asymmetry exceeds the driving frequency and strength. Moreover, the nonlinear QSR shows a noise-induced suppression of higher harmonics and a characteristic phase shift.