No, they didn't. They only had the century locked down, with the "In the 23rd century..." opening text. Star Trek didn't use a specific calendar year for the events of an episode or movie until TNG produced "The Neutral Zone." (And the stated date of 2364 likely only made it into the finished episode because of the 1988 Writer's Strike.)
Although no specific date was given, prior to airing TNG was promoted as being "78 years after the original Enterprise", and it was definitely in the 24th century. So, that provides a range of possible years, albeit a rather large one.
TWOK used 2283 as a date for the Romulan Ale that McCoy gave to Kirk, but it didn't provide any context for it outside of "Well, it takes this stuff a while to ferment." We aren't even told if the year is from a human or Romulan dating system.
It makes more sense for 2283 to be a stardate, actually. That puts it's production back in the original series, which had been established as 15 years earlier (and the stardate during Space Seed is actually 3100 or so, making the ale older than that).