Could injuries to the testicle cause fertility problems?

Maybe. Testicular trauma, if severe, can rupture the testicle and cause impairment of sperm production but this is very rare. If a man has just had typical playground testicle injuries, it's unlikely to explain fertility problems and he should get a more thorough evaluation.
Yes. The immune system usually is not in contact with the testicle, and thus trauma to the testis can have your immune system be exposed to sperm for the 1st time and you can develop antibodies against your own sperm.