Ist der Satz grammatikalisch korrekt: „Könnte es etwas damit zu tun gehabt haben?“?

In any case it is bad expression, not good German. Today, some things are considered correct, which was wrong a few years ago. Thus, the question of linguistic entitlement rather than the question of right or wrong. Better would be:
“Can it be that it had to do with it?”

Question plus conjunction is, however, a logical doubling. You occasionally formulate so cautiously to not get close to someone. It’s a little blurred.
“Does it have to do with it?”
This is just as logical.

Could be conjunctive, had is perfect, and having to do is an infinitive. That’s what didn’t belong together. 😉