In this paper, we combine two new techniques: non-orthogonal multiple access (NOMA) and full-duplex (FD) in a downlink of wireless cooperative relay communication system with two end users. The NOMA approach is applied to both the source and the relay. Whereas, the FD is only applied to the relay. The system performance is investigated in the case of vehicle-to-vehicle (V2V) communication. The self-interference (SI) due to FD protocol is taken into consideration, and then the outage probability (OP) and ergodic capacity (EC) of the considered FD-NOMA-V2V system are derived over the double Rayleigh fading channels. Furthermore, an impact of distance between the vehicles and a path loss exponent on the system performance is discussed. The numerical results show that the system performance strongly decreases in comparison with that over the Rayleigh fading channel, and the SI lets the performance of the considered system decrease significantly. Due to the FD mode and NOMA scheme, the system performance in terms of OP and EC reaches the floor at the high signal-to-noise ratio (SNR) region. Based on the distance between the vehicles, the path loss exponent and the residual self-interference (RSI), the power allocation coefficients is decided in the sense of similar performance of both users in the cooperative FD-NOMA-V2V system. Additionally, we can choose locations of the relay to achieve the best performance of the considered system. Finally, the perfect match between the theoretical and simulation results validates our proposed method.