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As the wind power generation becomes increasingly popular, the conventional inverter control methods may lead to power grid instability due to the reduction of inertia and susceptibility to parameter variations. In order to address this problem, a virtual synchronous generation (VSG) technology is combined with the quasi Z-source inverter (qZSI), which has the built-in boost function. To begin with, a finite switch sequence control strategy and an unconstrained optimization model predictive control (MPC) strategy are introduced to achieve multi-objective optimal control of the qZSI-VSG system. The average switching frequency of the inverter is reduced, the ability of the system to mitigate fluctuations in wind turbine output is enhanced, and the current harmonics are minimized. In addition, the computational complexity of the algorithm is simplified by eliminating the need for weight coefficients to be directly added to the cost function. Finally, the correctness and effectiveness of the proposed control strategy are verified by theoretical analysis and experimental results. Overall, this approach offers a promising solution to the challenges posed by the increasing integration of wind power into the power grid.