It isn't consistent with the comics...
... not the stories that take place within the prime, Earth-616 continuity, at least. In that continuity, the Avengers team with Thor, Hulk, Iron Man, etc wasn't actually the first Avengers team, canonically speaking.
A later story established that the first Avengers team was formed in 1959 (Marvel Universe-time), before Carol Danvers was even born, never mind in the Air Force. (We know she was in the Air Force prior to the formation of the modern Avengers, since she and Ben Grimm, AKA the Thing, flew on a mission together when he was still a pilot himself, before either the Fantastic Four or the modern Avengers had formed.)
Nick Fury (the original version) was involved with the formation of the 1959 Avengers, however, he didn't come up with the name; he was told what the name would be by his superiors, and instructed to recruit and lead the team. The dialogue indicates that the orders, and possibly the name of the team itself, came directly from the US President (which would've been President Eisenhower, assuming Marvel Universe history aligns with real world history in that regard).
New Avengers Vol 2 #9
Avengers 1959 Vol 1 #5
Also, as noted in Ross Thompson's answer, when the comicbook version of Carol was in the Air Force, her call sign was apparently "Cheeseburger", not "Avenger".
Ms. Marvel Vol 2 #32