Metasurfaces are a key photonic platform to manipulate classical light using sub-wavelength structures with designer optical response. Static metasurfaces have recently entered the realm of quantum photonics, showing their ability to tailor nonclassical states of light. We introduce the concept of space-time quantum metasurfaces for dynamical control of quantum light. We provide illustrative examples of the impact of spatio-temporally modulated metasurfaces in quantum photonics, including the creation of frequency-spin-path hyperentanglement on a single photon and the realization of space-time asymmetry at the deepest level of the quantum vacuum. Photonic platforms based on the space-time quantum metasurface concept have the potential to enable novel functionalities, such as on-demand entanglement generation for quantum communications, nonreciprocal photon propagation for free-space quantum isolation, and reconfigurable quantum imaging and sensing.