In 2293, the Kongo was assigned a Neutral Zone patrol mission. The ship was named as eighth on a Starship Mission Assignments chart on the USS Enterprise-A bridge. (Star Trek VI: The Undiscovered Country)
The name Kongo and its registry number were not from dialogue, but only derived from sheet four of the briefing charts background artwork for Operation Retrieve in The Undiscovered Country. However that particular page was not featured in any of the releases of the movie, though the preceding "Starship Mission Assignments" okudagram, on which the sheet was based, was, albeit it mostly illegible, at the communications station on the bridge. The registry originated from Franz Joseph's reference book Star Trek Star Fleet Technical Manual, which listed the Kongo – with the NCC-1710 registry – as a Constitution-class heavy cruiser. Although not considered as canon the Technical Manual was used for the references in it, as it had been in the first three Star Trek films, and which was for this case confirmed by Michael Okuda, who created both the "Mission Assignments" and "Operation Retrieve" briefing charts. 
The name Kongo though, had already been envisioned by the producers of Star Trek: The Original Series when they composed a list of fourteen ships at the start of its second season, belonging to the Constitution-class, then still referred to as "Starship-class" by them and including the Kongo. Producer Robert Justman remarked on a memo of an earlier draft of the list, "I think there would be several other candidates, such as Saratoga and perhaps another English carrier, a French carrier, a Russian carrier and certainly a Japanese carrier." (The Making of Star Trek, pp. 164-165) Though there had been plenty of historical Japanese aircraft carriers to choose from, the historical Kongo was in actuality a Japanese battleship of the World War I-II era. However, along with the Constitution, it was the only one of these ships that was neither seen nor named in the course of the series and its configuration was never canonically established.
In his influential 1973 The Case of Jonathan Doe Starship article, studio model vendor, but then fan, Greg Jein also postulated the Kongo to be Constitution-class, but in his case endowed with the registry "NCC-1732". Mike Okuda has since then fully endorsed Jein's postulations, employing them wherever he could in canon – most notably in his reference works, but more specifically in the remastered series – , but became aware of his friend's article too late to alter the Kongo registry, as he had already employed the Joseph-derived one. As a result, Kongo together with the USS Potemkin for the same reasons, are the only Constitution-class vessels of the original fourteen to sport Joseph-derived registries. As stated, no opportunity has been afforded to Okuda to retcon Kongo with the Jein registry, and due to its absence in canon (outside the mission assignment chart that is, illegible though it is) Kongo has not been acknowledged in any of the later officially licensed reference works, contrary to the "USS Constitution NCC-1700" which has as the class vessel, despite the fact that it too has not been confirmed as such in canon.
According to the unseen page four of the "Operation Retrieve" briefing charts, the Kongo was commanded by N. Rodis, and the ship was located in Sector 21803. The ship's captain was named after the sixth movie's art director Nilo Rodis-Jamero.