Yes, it is okay for characters to recognize elements of the Star Trek universe, up to and including any canonical Star Trek characters, within reason. Players are disallowed from telling those characters that they or their world is fictional, or chatting about their personal details (such as events that will happen to them in the future), unless all players involved have given express permission to do so (this would include mention of any active third-party character not involved in the thread).

Inner-monologues and/or narration should be safe enough, but if you're ever unsure about a player's preferences, it is always best to err on the side of caution. Having a permissions post available (example: here) with a section about canon-puncturing and fourth-walling where others may comment may be a good way of keeping track.