Collective plasmon modes, riding on top of drifting electrons, acquire a fascinating nonreciprocal dispersion characterized by omega(p)(q) not equal omega(p)(-q). The classical plasmonic Doppler shift arises from the polarization of the Fermi surface due to the applied DC bias voltage. Here, we predict an additional quantum contribution to the plasmonic Doppler shift originating from the quantum metric of the Bloch wavefunction. We systematically compare the classical and quantum corrections to the Doppler shifts by investigating the drift-induced nonreciprocal plasmon dispersion in graphene and in twisted bilayer graphene. We show that the quantum plasmonic Doppler shift dominates in moire systems at large wave vectors, yielding plasmonic nonreciprocity up to 20% in twisted bilayer graphene. Our findings highlight the significance of the quantum corrections to plasmonic Doppler shift in moire systems and motivate the design of innovative nonreciprocal photonic devices with potential technological implications.