The word "quite" is an adverb that may mean--entirely, wholly, really, very...
I'm quite hungry. Is there anything to eat?
We hadn't eaten since yesterday so we were getting quite hungry.
I'm quite sure about that.
It may mean --somewhat, rather, fairly but not very.
After eating the first Big Mac, I was still quite hungry, but couldn't manage to finish a second one.
Just looking at a picture of the typical British breakfast can make you feel quite hungry.
He was quite ill, which is why he stayed home.
Since the two meanings are somewhat opposed you need to rely on context for an exact understanding of "quite".
On the other hand " I am very hungry." always has the meaning of "extremely hungry" as Jura explained.