Nick Fury: I know a renegade soldier when I see one. Never occurred to me that one might come from above.
Carol Danvers: Higher, further, faster, baby.
Carol Danvers: Did you have a rough day, Agent Fury?
Nick Fury: It was cool, you know? Had a space invasion, big car chase. Got to watch an alien autopsy. Typical 9-to-5.
Carol Danvers: So you saw one?
Nick Fury: I was never one to believe in aliens, but I can't unsee that.
Carol Danvers: Has a Skrull ever simmed you?
Korath: Once... it was deeply disturbing.
Carol Danvers: Why?
Korath: Because I stared into the face of my mortal enemy and the face staring back was my own.
Talos: This is war. My hands are filthy from it too. But we're here now. You found my family, this is just the beginning.
Nick Fury: So... you're not from around here.
Carol Danvers: It's hard to explain.
Carol Danvers: I've been fighting with one arm tied behind my back. But what happens when I'm finally set free?
Carol Danvers: I'm not gonna fight your war. I'm gonna end it.
Nick Fury: You know anything about a lady blowing a hole through the roof of that Blockbuster over there? Witness says she was dressed for laser tag.
Nick Fury: Coulson. New guy. Guess he doesn't hate me yet.
Maria Rambeau: You call me 'young lady' again, I'll shove my foot up somewhere it's not supposed to be.
Talos: Am I supposed to guess where that is?
Carol Danvers: Your ass!
Carol Danvers: I have nothing to prove to you.
Carol Danvers: Nicholas Joseph Fury. You have three names?
Nick Fury: Everybody calls me Fury. Not Nicholas, not Joseph, not Nick. Just Fury.
Carol Danvers: What does your mom call you?
Nick Fury: Fury.
Carol Danvers: What do you call her?
Nick Fury: Fury.
Carol Danvers: What about your kids?
Nick Fury: If I have them, they'll call me Fury.
Answer: Howard Stark relinquished custody of it to her. For a full history of the Tesseract see here: https://marvelcinematicuniverse.fandom.com/wiki/Tesseract.