Not sure about other alien languages off the top of my head, but I believe the first instance of a human speaking Klingon was Kirk saying the words "Maltz... activate beam!" in that language in Star Trek III: The Search for Spock (1984).
At the time, Kirk had just killed Commander Kruge on the Genesis planet, and he spoke those words into a Klingon communicator in order to trick a Klingon officer named Maltz into beaming him and Spock up to Kruge's vessel. Kruge used his own communicator to give the same order to Maltz in Kirk's presence a few minutes earlier, so Kirk likely mimicked what he heard rather than having any prior knowledge of the words.
You can view the scene where Kirk speaks those words at around the 3:48 mark in the video clip below. Kruge speaks the same words at around the 0:33 mark:
Star Trek III: The Search for Spock (1984)
To add some background, a linguist named Marc Okrand is credited by multiple sources (including StarTrek.com) as being the real world creator of the Klingon language.
In an interview with StarTrek.com, he stated that Klingon was first spoken at the beginning of Star Trek: The Motion Picture (1979):
STARTREK.COM: For Star Trek III: The Search for Spock, you basically created the Klingon language. What went in to doing so?
MARC OKRAND: I started by looking at Star Trek: The Motion Picture, because that’s where Klingon was actually first spoken. There are perhaps a half dozen lines in Klingon with subtitles at the beginning of the film.
Qapla' -- Klingon Language Creator Marc Okrand, Part 1
The dialogue in question was spoken by a Klingon Captain (played by Mark Lenard) in the opening scene of the film:
Star Trek: The Motion Picture (1979)
Okrand was asked to flesh the language out for Star Trek III: The Search for Spock (1984). According to him, Lenard told him that the Klingon lines in the first film were devised by James Doohan. Okrand used those lines as a starting point and expanded the language from there:
MARC OKRAND: I wrote down those lines as best I could, made a list of the sounds in the words and figured out what a legitimate syllable was. That was the beginning. All of the sounds and all of the syllables and, for that matter, all of the phrases in the first movie formed the skeleton of what I was to build. When I was looking at the first film, I didn’t know who made up those lines. When I met Mark Lenard, the actor who spoke them in that film, he told me that they were devised by James Doohan. So James Doohan actually originated Klingon. I came along and fleshed it out.
In this process, I had four things in mind as far as the sounds go: (1) the language had to include all the sounds in the first film; (2) the language had to have non-English sounds since it was to be alien; (3) the language had to be guttural, since the script for Star Trek III explicitly referred to Klingon as a guttural language; and (4) it had to be learnable and pronounceable by the English-speaking actors, so it contains many ordinary English sounds in addition to the more exotic sounds. I also tried to make the grammar non-English-like. The vocabulary was easy – I made up only what was needed for the film. If a word didn’t come up, I didn’t come up with a Klingon equivalent. The same went for the grammar – if a particular construction or grammatical element, say a pronoun, wasn’t needed for the film, I didn’t make it up. Later on, I added lots of vocabulary and grammar – stuff not in the film. But initially, the script drove what was made up.
Qapla' -- Klingon Language Creator Marc Okrand, Part 1
