Electron quantum transport is theoretically studied for finite-size armchair graphene nanoribbons biased within source and drain metallic electrodes, using an extended-Hückel-based Green's function coupled to a three-dimensional Poisson solver. The analysis evidences dynamic nonequilibrium electron charging phenomena that can affect the conduction mechanism by provoking electronic structure alterations. The origin of such process can be traced in a tracking relationship between the device's local density of states and the electrochemical potentials of the contacts. Such effect has no equivalent in the semiclassical limit.