It doesn't appear likely that any of the people aboard would actually know how to speak Klingon. How many USN personnel spoke Russian during the Cold War? And that's a language they would have had available in libraries everywhere, rather than an alien one.
In ST3, it's something of a plot point that Kirk doesn't know any Klingonese, and only manages to get Maltz to beam him up by imitating a phrase he overheard from Kruge...
In was Archer, not Kirk, who had a linguist as part of the bridge crew. Well, and nuPike, too; apparently, nuKirk has one now as well.