Updated December 6, 2018: The official description for the Captain Marvel YA novel Starforce on the Rise confirms that Carol Danvers goes by the name "Vers" among the Kree - presumably resulting from the broken dogtags seen in the most recent trailer, in which that part of her name appears to be cut off.
Here's the full synopsis for Starforce on the Rise: "Before Carol Danvers was Captain Marvel, she was Vers, a key member of the elite Kree Starforce warriors who wage their war against the equally formidable Skrull aliens. This middle-grade novel will focus on key adventure of Vers during her time with Starforce, setting up the perfect entry point for readers to Marvel Studios’ Captain Marvel, in spring 2019."
Original Story: According to Marvel Studios, 2019's Captain Marvel begins with Carol Danvers living among the Kree as an officer in Starforce, but apparently they don't call her by Carol or by 'Captain Marvel' (yet); instead, apparently it's 'Vers.'
That factoid was apparently revealed as part of Funko',s just-announced line of Captain Marvel merchandise. Danvers' green-costume get-up - her Starforce uniform - carries the 'Vers' name, while the latter red-and-blue uniforms are dubbed Captain Marvel. There are also 'Carol Danvers' figures depicting her as an Air Force pilot and as a civilian riding a motorcycle. If what Funko lays out here is in line with the movie, Carol is first known as 'Vers' to the Kree but later becomes 'Captain Marvel' after a costume change.
In comic books, Kree individually typically have a hyphenated name - Mar-Vell, Yon-Rogg, etc. - with the second half acting as their surname. So in this case, Carol's full Kree name could presumably be 'Dan-Vers.'
It hasn't been disclosed if she has a rank in the Kree military, be it Captain, Corporal, Private, or what not.
Captain Marvel is scheduled to open in theaters March 8, 2019.