The bidirectional quasi-Z-source inverter (qZSI) driven by permanent magnet synchronous motor (PMSM) not only has the advantages of both the Z-source inverter and the PMSM, but also realizes the bidirectional flow of energy. According to the characteristics of the system, a fast vector selection finite control set-model predictive current control (FCS-MPCC) strategy is proposed. The reference value of inductance current is generated by controlling the DC link voltage of quasi-Z-source network with the electromagnetic power feedforward value of PMSM. The reference motor current is generated by the speed closed-loop control. By predicting the inductor current value and calculating the inductor current subcost function in the shoot-through (ST) case and non-shoot-through (NST) case, either the ST case or NST case is chosen to realize the control of DC link voltage. In the NST case, combined with the space voltage vector pulse width modulation strategy, only the four vectors of the sector where the target voltage vector is located participate in the PMSM stator current prediction and cost function calculation to select the optimal switching state. It can reduce the amount of online calculation while controlling the speed of the PMSM. The simulation results show that the proposed control strategy can realize the boost control of the bidirectional qZSI and the speed control of the PMSM under traction or braking conditions, and the system has good steadystate and dynamic performance.