In this paper, we consider a two-cell interfering multiple-input multiple-output multiple access channel (MIMOMAC). We first investigate the multiplexing gain of such channel when users have perfect channel state information at the transmitter (CSIT) by exploiting an interference alignment scheme. In addition, we propose a quantized transmit beamforming vector feedback method for the interference alignment in the limited feedback system. On the basis of the proposed feedback framework, we analyze the rate gap loss and it is shown that in order to keep the same multiplexing gain with the case of perfect CSIT, the number of feedback bits per receiver should be increased linearly with the SNR in dB scale. Throughout the simulation results, it is shown that the sum-rate performance coincides with the derived results.