In late 2366, the USS Chekov was one of 40 starships mobilized at Wolf 359 to intercept a Borg cube en route to Sector 001. The Chekov engaged the Borg at the subsequent battle where it was destroyed, along with the rest of the fleet. (TNG: "The Best of Both Worlds, Part II")
The name and registry came from the studio model. The Chekov was classified as a Springfield-class starship in an internal list of starships, dated September 26, 1990.  This information about the ship later appeared in the Star Trek Encyclopedia (4th ed., vol. 2, p. 327).
The Star Trek Encyclopedia (3rd ed., p. 471) stated the ship was "named for the noted Russian space explorer."
The ship's name was an homage to the original series character Pavel Chekov. This starship (named Chekhov, after the Russian writer, in the episode script) was originally to be mentioned by name in dialogue in "The Best of Both Worlds", but the writers ultimately felt that the debris field that remained of Humanity's apparent last stand was no place for such a "cute" reference, and the USS Tolstoy was dubbed in during post-production to replace it.
Its registry, according to the model featured at Ex Astris Scientia, was NCC-57302. The Star Trek Encyclopedia (4th ed., vol. 2, p. 327), on the other hand, listed the registry as NCC-53702, which could be a typographical error as it appeared to differ from the studio model used for the visual effects of the battlefield. Photographs of this model were part of Michael Okuda's art department slide show given at conventions throughout the 1990s.