The problem doesn't have anything to do with GSX, and it's caused by the fact the MJC Q400 has a non standard flight model using its own internal simulation variables, so GSX cannot detect if the engines are running like it can with every other airplane.
When it's required for GSX to detect if engines are turned off (in order to activate services when parked), users of the MJC Q400 plane have suggested use the CTRL+Shift+F1 command, which by default shut down the fuel valves, turning off the default engine simulation which is not used by the airplane, but it's required by GSX.
So, you might try the opposite command ( CTRL+Shift+F4 ), and see if it works in reverse to help GSX detect engines on with that airplane.
Otherwise, just not use the engine start before pushback option with the MJC Q400.