In Captain Marvel, we see that Captain Marvel's callsign is Avenger, and we find out that the Avengers Initiative was named after her. So, do the Avengers ever find out that they were named after her callsign?
Not that we see on screen or off it so far. Note that this was made up for Captain Marvel and so films before that wouldn't have had that information to put into them. After that we've only had two major films set after Captain Marvel returned to Earth: Avengers: Endgame and Spider-Man: Far From Home. Neither film, per my knowledge, mentions the origins of the name. I've also quickly checked the various relevant Prelude comics tie ins and it doesn't appear to be mentioned there.
I will note though that Captain Marvel doesn't necessarily know where the name Avengers came came from. She might guess it came from her but unless she was told off screen she won't know. Remember that Nick came up with the name after Carol had already left Earth again and she doesn't return, as far as we know, until Endgame. That means for people to know where the name has come from Fury would have had to tell them. That doesn't necessarily seem to be the thing he would do, certainly not after Carol had already become a member and had been for a few years.