The first mention of Officers' Quarters was in "Where No Man Has Gone Before" where a directional sign was seen with this label. Through the run of the series, there were a handful of other directional signs for these quarters. The first and only dialog reference to these quarters was in the first movie.
In these quarters, there was a door that lead to an unseen room. In a deleted scene from "The Corbomite Maneuver", this was revealed to be a closet. 
In the script for the first movie, the Officers' Quarters were located on Deck 4.