External quantum efficiency (EQE) is the ratio of electrons out to photons in. It is wavelength dependant. For most solar cell technologies and illumination levels the quantum efficiency is assumed to be independent of illumination.
A lot of factor go in to determining the EQE of a solar cell, the dominant contribution is from the materials used and device structure. Two crucial parameters for solar cells are the absorption coefficient and minority carrier diffusion length. These directly influence the EQE. If these processes are dependent on illumination intensity then the EQE will change.
For solar intensities, absorption can be assumed to be a constant. Although under extreme illumination levels (using lasers) photo-filling can reduce the absorption coefficient because of reduction in available occupied states in the valence band and unoccupied states in the balance band. This is not really relevant to solar cells under normal operating conditions.
If the minority carrier diffusion length changes, so too will the EQE. Moreover, if recombination rate increases then EQE will decrease. Again, this does not really happen at solar intensities. However, under concentrated solar light, around 300 suns Auger recombination begins to limit the performance of Silicon solar cells.
So to first approximation, EQE can be considered independent of illumination level, for most solar cell technologies and operation conditions. Although it does have a weak dependence because it is linked to recommendation rate of minority carriers in the device and that can have a dependency on illumination level.