Supergirl is named Kara Danvers (by adoption), and in the latest TV-show iteration, works with her sister Alex Danvers.
Captain Marvel (upcoming movie version) is named Carol Danvers.
Any connection, relation, shout-out, anything like that?
It is always difficult to say whether such similarities are just coincidence, or if there was some intended reference. In this case, however, it is most likely just a coincidence.
When Supergirl was first introduced in 1959, her original secret identity was Linda Lee. She was later (in 1961) adopted by Fred and Edna Danvers and was from then on known as Linda Lee Danvers.
Carol Danvers was first introduced in 1967 in Marvel Super-Heroes #13 in the role of Security Chief of a restricted military base, where Danvers meets Dr. Walter Lawson, the human alias of alien Kree hero Captain Marvel (introduced in the same issue). She didn't get any superpowers until 10 years later, in Ms. Marvel #1 (1977), and as far as I can see, she didn't appear in any comics between Captain Marvel #18 in 1969 (where she sustained severe injuries) and Ms. Marvel #1. It seems that originally, she was not intended to be a superheroine, and hence it is unlikely that she is named for Supergirl's alias.