Not quite. A quick primer on the various Marvels in Marvel Comics:
First, you have to understand that Captain Marvel is all about keeping a copyright going. Captain Marvel in the 1950s was the character that we know today as Shazam – he was owned by Fawcett Comics, but they let the trademark to the name lapse in the 60s when they lost a court case against DC comics. Marvel Comics grabbed up the name and published their first Captain Marvel story in 1968. The name has been used by a slew of characters, but Marvel publishes a Captain Marvel story every few years to keep the trademark valid.
The first Captain Marvel owned by Marvel was an alien Kree soldier named “Mar-Vell” who came to Earth to fight evil, etc., taking on the name “Captain Marvel”.
Carol Danvers is caught in an explosion with Mar-Vell, and gains some of his powers. She takes on the name “Ms. Marvel” in 1977.
The first Captain Marvel (the alien) would die of cancer in 1982 in a big comic-book event… and remarkably has mostly stayed pretty dead… though of course as noted above, Marvel Comics is obliged to roll out a new Captain Marvel every few years to keep the name going.
The character of Monica Rambeau was actually the first female Captain Marvel, back in 1985, before changing her name to “Photon”.
In the 90s and 2000s, the moniker “Captain Marvel” has been borne by Mar-Vell’s son/clone, his son/clone’s sister, a brainwashed Skrull, and another Kree soldier.
Meanwhile, Ms. Marvel (Carol Danvers) was a mainstay in The Avengers, still bearing the name “Ms. Marvel”. The character kind of disappears for a while after a seriously weird story-line in Avengers #200, and in 1981, the X-Men villain “Rogue” steals her powers. The character goes through some changes and ends up being named “Binary” for a while.
There were a couple minor characters who bore the name briefly (again, to protect the trademark) as Carol Danvers cycled through various power-sets and names.
In 2012, Carol Danvers takes on the “Captain Marvel” title and has held it ever since.
In 2014, the character of Kamala Khan – a Pakistani-American teen – took the name “Ms. Marvel” since she idolized Danvers and knew the name wasn’t being used.
And I think that’s up-to-date. Thank you, Wikipedia.