BJT Push-Pull explained
I think you are right. When a Voltage is applied to the gate of Q1, a current can flow from VCC which will charge the gate capacitor of Q3 which will raise it the gates Voltage to VCC. The time this will take is depending on the value of the parasitic gate capacitance and the gate resistor R2.
Addition:
When the PWM signal goes low again, the lower BJT will start conducting pulling the MOSFETs gate to ground by discharging its capacitor.