This paper proposes a new four-quadrant adjustable speed drive (ASD) system based on an induction motor fed by a three-phase quasi-Z-source direct matrix converter (QZSDMC). The quasi-Z-source (QZS) network is used to overcome the limitation of the voltage gain of the traditional direct matrix converter (DMC). The QZSDMC can provide a buck-boost operation by controlling the shoot-through (ST) duty ratio. In addition, this paper presents a four-quadrant speed control method based on vector control technique which is able to control the motor speed from zero to the rated speed with the full load torque during motoring and regenerating operation modes. The simulation results validate the proposed QZSDMC based adjustable speed drive system. The proposed ASD system overcomes the voltage gain limitation of the traditional direct matrix converter with improved efficiency and higher readability compared to the traditional back-to-back converter configuration.