We mathematically derive an upper bound of bit-error rate (BER) of uplink non-orthogonal multiple access (NOMA) systems with quadrature phase shift keying (QPSK) modulation in fading channels. In particular, we exploit the joint maximum-likelihood (ML) detector at a base station (BS) with multiple antennas since it results in the optimal BER performance of a super-imposed QPSK modulated symbols from multiple users. In particular, we obtain a closed-form integral of Q-function with Erlang distributed random variable to derive the BER. We also obtain diversity order of the uplink NOMA systems. Through extensive computer simulations, we validate that our analytical results match well with the simulation results especially in high signal-to-noise ratio (SNR) regime.