The recent synthesis of a two-dimensional quasi-hexagonal-phase monolayer network of C60 molecules, known as qHPC60, holds significant promise for future semiconductor applications. However, the mechanism behind charge transport in these networks remains unknown. In this study, we developed a Holstein-Peierls Hamiltonian model to investigate charge transport in qHPC60, incorporating both local and non-local electron-phonon couplings. Our computational approach involved identifying suitable semi-empirical parameters to realize the formation of stable polarons in this material. The results unveiled the formation of stable large polarons as the primary carriers in the charge transport throughout qHPC60. To explore polaron transport properties, we conducted dynamic simulations within the picosecond time scale while subjecting the system to an external electric field. Our analysis emphasized the substantial influence of anisotropy on shaping mobile polarons, with an anisotropy coefficient of at least 50%. The polarons exhibited velocities within the acoustic regime ranging from 0.5-1.5 nm ps-1. While these velocities are comparable to those observed in high-end organic molecular crystals, they are considerably lower than those in graphene and conducting polymers. With qHPC60 possessing a semiconducting band gap of approximately 1.6 eV, our findings shed light on its potential application in flat electronics, overcoming the null-gap predicament of graphene.