Multi-carrier power-domain non-orthogonal multiple access (MC-PD-NOMA) has been considered a key technology for next-generation wireless networks. In MC-PD-NOMA, the available channel bandwidth is divided into several sub-carriers in order to serve the multiple users in each sub-carrier. In this work, to reduce the number of successive interference cancellation (SIC) operations, detection delay, SIC complexity and to enhance the SIC stability of the MC-PD-NOMA, a system, namely, MC-quadrature-NOMA (MC-Q-NOMA), is proposed. Furthermore, the proposed MC-Q-NOMA follows the essential upconversion architecture to provide orthogonality in the pass-band. Moreover, a novel analytical framework is introduced to evaluate analytical expressions of average sum-rate (ASR), outage probability (OP), and diversity order for the proposed system under the generalized $\alpha -\mu$ channel for perfect and imperfect SICs. In addition, the proposed framework is utilized to determine the analytical expression of ASR, OP, and diversity order for MC-PD-NOMA. Comparative analysis indicates that under imperfect SIC, MC-Q-NOMA outperforms the MC-PD-NOMA due to the lesser number of SICs. However, under perfect SIC, MC-Q-NOMA demonstrates comparable performance with the MC-PD-NOMA. Furthermore, the analytical expression of the average bit-error-rate is evaluated for the MC-Q-NOMA. Numerical results demonstrate that the error performance of MC-Q-NOMA is superior to MC-PD-NOMA for all signal-to-noise ratio values.