What is the equivalent of "quench" when speaking of hunger? Is it appropriate to say you quenched your hunger?
Sate, "To satisfy; fill up" is the usual term.
In etymology, quench and sate are somewhat parallel: quench derives via an Old English word from a Proto-Germanic word, while sate derives via a Middle English word from an Old English word from a West Germanic word.
Note, sate came into use half-a-century before satiate, the latter directly from Latin satiatus, pp. of satiare, "fill full, satisfy", and the former from M.E. saden ("become satiated") as an alteration under the influence of L. satiare.
The formal answer is that you satiate a hunger, and you quench a thirst - as @jwpat7. The problem is that both of these words are used for other drives or needs, and they have differing meanings for the form of resolving the drive.
Quench tends to mean pour water on, not necessarily to satisfy the need, as much as to cool it - of course, when it comes to thirst, the water or liquid will satisfy it too. The image I always have is of being in a waterfall, where you can drink as much as you want, and there is still plenty, but you are going to get very wet.
Satiate can be used for a more "animal" satisfaction, where the person takes what they need, and then stops. It is far more agressive, far more a lion finding its prey, killing it, and when satiated, leaving the remnants for other beasts.
So while they are parallels, they also convey different senses.
You could stanch or stay your hunger: take the edge off it with a relatively small meal (or a large one, used ironically): or fulfil, repair or other generalised words implying filling in a lack. ("a hunger that it took forty minutes solid work to repair", Jerome K Jerome, Three Men on the Bummel.)
Assuage can be, and often has been, very effectively used with hunger.
to make milder or less severe; alleviate; ease; mitigate:
to assuage one's grief;
to assuage one's pain.
to appease; satisfy; sate:
to assuage one's hunger.
to soothe, calm, or mollify:
to assuage his fears;
to assuage her anger.
I'd go with "stave off hunger".
stave something off
to delay or postpone something unwanted, such as hunger, foreclosure, death, etc. (See also stave someone or something off.) He could stave his thirst off no longer. Despite the enemy sentries, he made a dash for the stream. The lost hiker could not stave off her hunger any longer.
I see hunger and thirst as things that can be temporarily avoided, but not permanently disposed of. The definition does also refer to thirst, but I've heard "stave off" referred to hunger more than thirst in my experience, as evidenced by this Google Ngram:
Another term which implies a similar meaning, but which is used less often is "to keep hunger at bay".
When hunger no longer exists, I'd probably say "I'm full" rather than a statement indicating that my sense of hunger has dissipated. The way I see it, a feeling of fullness replaces hunger much more than a feeling of satiation replaces thirst, so there isn't a direct alternate to the word "quench" in relation to hunger. The closest thing I can think of would be to "satisfy your hunger". Here's a Google Ngram comparing "quench your thirst" to "satisfy your hunger", as well as a comparison to their opposites of "quench your hunger" and "satisfy your thirst":
Comparing "stave off hunger" and "satisfy your hunger" ends up roughly equal in usage, at least according to the Google Ngram:
In the end, it's up to you as to what you wish to communicate.