Did King Solomon kill any men and if so who?

Yes, King Solomon ordered the execution of several men during his reign. Notably, he had Adonijah, his half-brother, killed after Adonijah attempted to claim the throne. Solomon also executed Joab, the former commander of David's army, for his past treachery and for supporting Adonijah. Additionally, he had Shimei killed for breaking his oath to not leave Jerusalem.