2nd what badwolf says, you’ve probably altered your form slightly from limping around when you were injured. Now, the “good” leg is pushing harder than the “bad” leg, and this imbalance is causing extra torque on one side. That’s why just one side hurts.
The more you squat and deadlift using this “impaired” technique, the worse the imbalance will become. You mentioned the quad injury, then the hamstring pull. That sound like at least several months of one of your legs lagging behind the other. One side of your body, including your lower back, obliques, abs is probably stronger than the other.
When you do the big moves, the strong side kinda takes over and the weak side works like an outrigger. You just need a few small rehab type moves to get your weak side functioning properly, then some single leg moves to strengthen it.
These problems are super common. I’m dealing with the same thing myself.
Here’s another thread about it
Here’s an article about training for symmetry