IMO there are are three main things that cause a back squat to be quad dominant. In random order.
1#- You are a short person, and have a small compact squat and you are able to keep your torso extremely vertical during your back squat placing the emphasis more on the quads, similar to a front squat.
2#- You have the ability to go deep but dont really know what you are doing and stop a bit above parallel. Almost all people squatting in this way will feel their quads primarily. I will also place at number 2 form, meaning, all the individual has to do is to learn how to do a deep squat, which is not too difficult, and they are fine. The latter does not happen very often.
3#- Most commonly, you lack the ability to hit your glutes and hamstrings during a deep squat. I am not talking about a powerlifting squat which is biomechanically a goodmorning/squat hybrid, but a normal moderate stance olympic squat, or something very similar (there are degrees).
Now by lacking the ability to hit your glutes and hamstrings, I mean you do not have the hip mobility (glute and hamstring flexibility) to get well below parallel and most times even to parallel because either your hamstrings and/or your glutes are tight to the point that they pull your pelvis under prematurely and when that happens you are going to lose the ability to create tension on the posterior chain.
Take home from this? Its probably not that you dont know how to get the glutes and hamstrings involved, it is that you CANT get them involved until you open them up. And a combination of dynamic and static stretching works the best.