There are likely a few reasons here so lets break them down.
For emergencies only
Carol says herself that Nick should only use the upgraded pager for emergencies, he doesn't appear to use it for the Chitauri invasion in The Avengers or Ultron in Avengers: Age of Ultron.
Carol Danvers: I upgraded it. Range should be, a couple galaxies, at least.
Nick Fury: What? You think I'm gonna crank call you?
Carol Danvers: For emergencies only. Okay?
This is also the reason why he doesn't appear to call her until Avengers: Infinity War, before then these threats were just Avengers level threats, Thanos was a Carol level threat.
Peter: And I know you probably have an answer for this. Why didn’t Nick Fury not use the pager until Infinity War? Because this big hole opened up in the sky and aliens attacked New York City.
Kevin Feige: Well, I’d say two things. One, she does say it’s gotta be a real emergency, right? Yeah.
Peter: Because that was just a city. I guess that’s not the world.
Kevin Feige: The other thing I’d say is how do you know he never hit it? How do we know he never pushed it before? We’ve never seen him push it before. That doesn’t mean he never did.
/Film, Why Nick Fury Didn’t Page Captain Marvel Before ‘Infinity War’, According To Kevin Feige
Which leads into the next point...
He may have called her and she couldn't come
He says she's "unavailable" which means he must know that for some reason. Whether that's because she told him when she left or he called and she told him, who knows? And even if he did call her, she's covering a large area as Earth isn't the only place with problems. Therefore, Earth probably wasn't her priority during that timespan of Spider-Man: Far From Home.
James "Rhodey" Rhodes: Hey, new girl, everyone here is about that superhero life. And if you don't mind my asking, where the hell have you been all this time?
Carol Danvers: There are a lot of other planets in the universe. And unfortunately, they didn't have you guys.
Natasha Romanoff: Carol, are we seeing you here next month?
Carol Danvers: Not likely.
Rocket: What, you gonna get another haircut?
Carol Danvers: Listen fur-face, I'm covering a lot of territory. The things that are happening on Earth, are happening everywhere, on thousands of planets.
Rocket: All right, all right, that's a good point. That's a good point.
Carol Danvers: So you might not see me for a long time.
It's worth noting that in the film that Fury we see,
isn't actually Fury, but Talos.
As such he might have had specific orders to not call Carol with whatever she was doing. In fact, it seems like he wasn't even supposed to call
considering his reaction in the scene when he does. As he doesn't even know what to say when people ask where the Avengers are it seems likely he, himself, is on a need to know basis and is just there temporarily. Even more so because his mission appears to be to fill in and give Peter the glasses.
In fact considering the relationship between this Fury and Carol it seems likely he may not have wanted to call her and instead rely on what he had. That's mostly why he says "Don't invoke her name.", not because of the power she has.
It is also worth noting that Beck has been manipulating "Fury" somewhat throughout this time. He's been making himself look very good as well as making the attacks occur near Peter so that he was the one Fury called upon as he was nearest. Whilst Beck can't really manipulate Fury into not calling the other Avengers he could try and make it seem like they were less needed by making himself look a lot better. When we get to London we even see Beck state that "No other Avengers are coming".
Along these lines it's worth noting that according to Jon Watts, the director, the whole thing probably wouldn't have worked if Fury had been there instead of Talos as he is more sceptical by his very nature.
“Nick Fury’s super power is essentially being skeptical,” Watts notes. “Suspicion is what makes Nick Fury, Nick Fury. That just helped make that part of the story make sense to me in a good way. Nick Fury was sort of taking a vacation of his own, which fits into the greater storyline of Peter’s vacation. So it all sort of came together in this one last moment.”
Polygon, Why Spider-Man: Far From Home’s director wanted a Sixth Sense-like, post-credit twist