The quantized Hamiltonian dynamics (QHD) theory provides a hierarchy of approximations to quantum dynamics in the Heisenberg representation. We apply the first-order QHD to study charge transport in molecular crystals and find that the obtained equations of motion coincide with the Ehrenfest theory, which is the most widely used mixed quantum-classical approach. Quantum initial conditions required for the QHD variables make the dynamics surpass Ehrenfest. Most importantly, the first-order QHD already captures the low-temperature regime of charge transport, as observed experimentally. We expect that simple extensions to higher-order QHDs can efficiently represent other quantum effects, such as phonon zero-point energy and loss of coherence in the electronic subsystem caused by phonons.