<p>This article investigates quasi-matrix projective synchronization (QMPSn) in quaternion-valued delayed fractional-order memristor-based Cohen–Grossberg neural networks (QDFMCGNNs). The quaternion-valued framework and synchronization concept are introduced, with a sufficient synchronization condition derived using a controlled error system that enables stable regulation of multidimensional neural states. The quaternion-valued formulation is practically relevant as it enables compact modeling of strongly coupled multi-component neural signals. The condition is developed using discrete time delays, Lyapunov theory, fractional-order differential inequalities, and fixed control strategies. By incorporating fractional-order dynamics, the proposed model captures memristive memory effects, enhancing adaptability to delays and nonlinearities, while a Lyapunov-based framework rigorously characterizes synchronization stability and convergence under nonlinear dynamics, time delays, and fractional-order effects. Moreover, synchronization error bounds are obtained, demonstrating provable robustness and predictable convergence with reduced conservatism, while supporting QMPSn under flexible projective configurations through systematic matrix design. A numerical simulation is provided to verify the effectiveness of the proposed results.</p>