Using dynamical Hartree-Fock mean-field theory, we study the high-harmonic generation (HHG) in the fullerene molecules C$_{60}$ and C$_{70}$ under strong pump wave driving. We consider a strong-field regime and show that the output harmonic radiation exhibits multiple plateaus, whose borders are defined by the molecular excitonic lines and cutoff energies within each plateau scale linearly with the field strength amplitude. In contrast to atomic cases for the fullerene molecule, with the increase of the pump wave photon energy the cutoff harmonic energy is increased. We also show that with the increase of the electron-electron interaction energy overall the HHG rate is suppressed. We demonstrate that the C$_{70}$ molecule shows richer HHG spectra and a stronger high-harmonic intensity than the C$_{60}$.
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