Is this sentence grammatically correct?
He picked up a quarrel.
If you mean that he's about to shoot a stubby arrow from a crossbow, then yes. If you mean that he tried to start a fight, the idiom is, "He picked a fight".
Is it syntactically correct? Yes, of course.
Is it semantically correct? Less clear. Was he a crossbowman who’s spilt his quarrels on the ground? Or was he a short-wave operator trying to tune in a distant channel of people bickering?
In either of those cases, sure. But if we’re talking about somebody being quarrelsome and starting one up, then no. One starts quarrels; one does not pick them up.