<p>This paper tackles the challenging issue of synchronization problem for quaternion-valued inertial neural networks (QVINNs) with mixed delays, encompassing both time-varying discrete and distributed delays. A hybrid control strategy combining sampled-data control and quantized control is proposed to integrates a semi-Markovian jump process to model random structural transitions, enabling a more flexible and realistic representation of stochastic switching behavior compared to conventional Markovian models. Leveraging a non-decomposition approach to retain the full algebraic structure of quaternion matrices, a rigorous synchronization error system is formulated. An advanced input delay approach and Lyapunov-Krasovskii functional(LKF) construction are utilized to derive new synchronization criteria in terms of quaternion-valued linear matrix inequalities (LMIs), which are then transformed into equivalent complex-valued LMIs for easier analysis. Furthermore, synchronization criteria under the event-triggered (ET) scheme are derived in a straightforward manner. Numerical simulations validate the effectiveness and performance of the proposed synchronization strategies, demonstrating their potential for practical applications.</p>