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The name Kongo and its registry number were not from dialog, but only derived from sheet four of the mission 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. 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 the "Operation Retrieve" mission 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.
According to the unseen page four of the "Operation Retrieve" mission 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.