Note the following:
We use the NEGF-DFT approach described in Brandbyge et. al. PRB 65, 165401, which is formally equivalent to PRB 63, 245407. This is now a well established methodology cited by more than 1800 authors. In this method the central region is modelled as an open system where the electrodes are charge reservoirs and charge can move in and out freely.
For metallic electrodes, the electric field will be zero in the electrode region of the central region, thus from gauss theorem the total charge inside the central region must be zero. Thus, for a finite bias calculation, the charge will not change with the bias but stay constant if you have included a sufficient part of the electrodes in the central region.
If you use an electro-static gate potential, i.e. include metallic regions, then you can have a charge which depend on the gate, source and drain voltage. The total charge in the central region will then be the negative of the gate charge, thus, this is a way to determine the charge on the gate electrode.