The Klingon Neutral Zone is first mentioned in TWOK. It's a mistake, basically. The de Forest Research company strongly recommended changing it to the Romulan Neutral Zone; they also recommended a reference to "Gamma Hydra" to keep the film in line with the television series. For some reason (perhaps to use the stock footage from TMP without comment), the Klingon Neutral Zone stayed, but Gamma Hydra was added to the dialogue.
The later films stuck to the idea.