What to do?
Even when squatting without weight, I just feel like the quads are doing 99.8% of the work. Lately I've been having some discomfort while squatting with higher weights. Is there a way to target specifically those muscles?
There are several things you can do, just know that in a raw squat (no squat suit or compression briefs) the glute involvement is primarily at the bottom, and the hamstrings are only moderately used. With a squat suit, the leverages change and loading the hamstrings is more important. That said, the glute and hamstring activity is still important--particularly for deadlifts.
Also, I recommend doing deadlifts if you don't already. It is the ultimate posterior chain movement.