If you've seen the movie a couple times then it becomes pretty clear that Nick Fury simply didn't know enough about Thanos until it was too late.
If you stretch the question a bit further, though, "Why didn't Fury call during the NYC attack or about Ultron?", then you have a much more interesting question.
One theory I haven't seen yet:
Imagine you've got a bank heist going on. Who would you call?
The police can probably handle it. They've got the tools and experience to sufficiently eliminate the threat.
If Iron Man is around, isn't too busy, and is willing to participate, then he could probably handle it better than the police. His technology and experience is superior. It'd be overkill to call him about every bank robbery, but you could be forgiven for doing so if you felt like the robbers were unusually dangerous. As Tony told Peter in Homecoming: "Forget the flying Vulture guy. There are people who handle this sort of thing."
Then there's Hulk... you don't call Hulk for a bank robbery because using him comes at great cost. Hulk is a destroyer, much like a bomb. You don't use him unless the situation warrants it. You would try to handle the situation with less raw, destructive power.
Perhaps the Captain Marvel character we're about to be introduced to comes with some costs. Perhaps she attracts the wrong crowd™, or her powers, once unleashed, are difficult to control. Maybe she just doesn't care that much about human life or collateral damage. Who knows... she might not have been much more useful in the NYC attack then that nuclear missile, which is why Fury chose not to call her.
But, if half of the universe is dying or already dead... well then the costs associated with using her may actually be warranted.