We investigate quantum stochastic resonance (QSR) in the periodically driven spin-boson system with Ohmic dissipation. For the special case α=1/2 of the Ohmic coupling strength we present exact QSR results. For arbitrary Ohmic coupling, the amplitude and the phase of the nonlinear response function vs temperature are studied in a vast regime of temperatures and frequencies. It encompasses a dissipative-dominated incoherent tunneling regime, occurring at ‘‘high’’ temperatures and adiabatic driving, as well as coherent tunneling regimes, reached either at ‘‘low’’ temperatures or/and nonadiabatic frequencies. As a general feature, a principal maximum appears only in parameter regimes where incoherent tunneling dominates over coherent transitions. Moreover, for adiabatic driving, the nonlinear QSR exhibits a noise-induced suppression of higher harmonics accompanied by a characteristic phase change.