A deadlock was a situation, typically one involving opposing parties, in which no progress could be made.
In 2267, Commodore Stone mentioned that he and Captain James T. Kirk had reached a deadlock as to whether to allow Spock to show them record tapes of Captain Christopher Pike's visit to Talos IV in 2254. However, Kirk disputed that it was a deadlock, since Pike himself, who had also been watching the recordings, hadn't been asked his opinion on the subject. When he was, Pike approved of the suggested continuation, so the observations of the Talos IV mission indeed proceeded. (TOS: "The Menagerie, Part I")
When the USS Voyager encountered the Q Continuum in 2372, Captain Kathryn Janeway used the term "deadlock" to refer to a dispute between Q and Quinn. The disagreement, on that occasion, was about whether Quinn would be forced to be taken away by Q or be granted asylum on Voyager. Janeway's solution to the deadlock was to hold a hearing onboard the starship, which resulted in Janeway ruling in Quinn's favor. (VOY: "Death Wish")