We study the small mass limit (or: the Smoluchowski–Kramers limit) of a class of quantum Brownian motions with inhomogeneous damping and diffusion. For Ohmic bath spectral density with a Lorentz–Drude cutoff, we derive the Heisenberg–Langevin equations for the particle’s observables using a quantum stochastic calculus approach. We set the mass of the particle to equal $$m = m_{0} \epsilon $$ m = m 0 ϵ , the reduced Planck constant to equal $$\hbar = \epsilon $$ ħ = ϵ and the cutoff frequency to equal $$\varLambda = E_{\varLambda }/\epsilon $$ Λ = E Λ / ϵ , where $$m_0$$ m 0 and $$E_{\varLambda }$$ E Λ are positive constants, so that the particle’s de Broglie wavelength and the largest energy scale of the bath are fixed as $$\epsilon \rightarrow 0$$ ϵ → 0 . We study the limit as $$\epsilon \rightarrow 0$$ ϵ → 0 of the rescaled model and derive a limiting equation for the (slow) particle’s position variable. We find that the limiting equation contains several drift correction terms, the quantum noise-induced drifts, including terms of purely quantum nature, with no classical counterparts.