This work presents a computational study of charge hopping dynamics along a one-dimensional chain with Gaussian site energy disorder and linearly coupled,quantum bath. Time-dependent square displacements are calculated directly from numerical solutions of Pauli master equations, for five different rate kernels: exact Fermi golden rule (FGR) rate expression, stationary phase interpolation (SPI) approximation, semiclassical (SC) approximation, classical Marcus rate expression, and Miller-Abrahams expression. All results demonstrate diffusive behavior in the steady state limit. The results based on the FGR rate expression show that the charge transport in the quantum bath can be much more sensitive to the disorder than the prediction from the classical Marcus expression. While the SPI approximation captures this general trend reasonably well, the SC approximation tends to be unreliable at both quantitative and qualitative levels and becomes even worse than the classical Marcus expression under certain conditions. These results offer useful guidance in the choice of approximate rate kernels for larger-scale simulations and also demonstrate significant but fragile positive effects of quantum environments on the charge hopping dynamics.