(written from a Production point of view)
The Star Trek: The Invasion of Klingon Empire is one of Tsukuda Hobby's SF Series games (item no. HG-009-4000, the last four digits indicating the price in Japanese yen). Produced in 1982 by the Japanese toy/game maker, and only released in Japan, it bears the additional subtitle "The Game of Space War in Star Trek".
This is a strategic hex- and counter-based ship-to-ship combat game (in the same vein as FASA's Star Trek: Strategic Operations Simulator game and the illustrious Star Fleet Battles games), given the four-part mounted map showing the United Federation of Planets, the Klingon Empire (or a portion thereof), the Romulan Empire (ditto), and a large amount of space between these. Production is clearly important, as there seem to be many rules sections covering Ship Factories. In the same vein, the ship designs (especially the Federation vessels) seem inspired by those in Star Fleet Battles (which were, in turn, inspired by those in Franz Joseph's Star Trek Star Fleet Technical Manual). The actual ship combat looks to take place on a smaller board, where the movement of ship squadrons takes place on the larger board.
The game definitely appears to have two sides, the Federation and the Klingons. It gives the appearance of allowing either two or four players – the map has a line down the middle which would split it reasonably enough for this purpose. Presumably two players split the Federation and the other side has a Klingon and a Romulan player.
The missions of the USS Enterprise are clearly of some importance to the game. The green lines on the map appear to chart these (many of the systems have names familiar from the original TV series), there is a rules section labeled "ST Game Enterprise Mission Chart," and one of the scenarios is called "Missions of Enterplize" (a bit of a typo from the base Japanese language used in the game). Other missions include "la Brandy de Camus" and a campaign game.