We utilize a nanoscale magnetic spin-valve structure to demonstrate that current-induced magnetization fluctuations at cryogenic temperatures result predominantly from the quantum fluctuations enhanced by the spin transfer effect. The demonstrated spin transfer due to quantum magnetization fluctuations is distinguished from the previously established current-induced effects by a nonsmooth piecewise-linear dependence of the fluctuation intensity on current. It can be driven not only by the directional flows of spin-polarized electrons, but also by their thermal motion and by scattering of unpolarized electrons. This effect is expected to remain non-negligible even at room temperature, and entails a ubiquitous inelastic contribution to spin-polarizing properties of magnetic interfaces.