The literal meaning of çok kalabalik is "full of fish": the fish in question being a particular Mediterranean species. çok is pronounced as "chock" and not "cock". Given the fact that etymologists when in doubt tend to pin things on the nautical world and the fact that there was a huge amount of trade between Istanbul and the West around the time that this phrase entered English, this seems quite plausible. If the phrase did enter English this way, it is unlikely that people would have written the fact down. So yes, possible.
(Stolen by Scunner)