I know for certain that a modern solid state game on which the head folds down onto the body will not fit in a Ford Fiesta (2007 5 door model). The height of the boot opening was not enough to allow the game into the back. My partner owned one, and we changed the car for a 2008 Honda Jazz which would take a game.
I presume that 4 Queens has a detachable head like most EMs which is held on by four bolts. Once the head is detached you should be able to slide the cabinet into the back and then open the back passenger door and place the head in next to the cabinet. The length of the cabinet may be an issue unless you have the front seats set fully forward.
The worst case is that you might need to make two trips to pick up the cabinet and head separately.