Main Canon
There's a throwaway line in the film to explain her decision to time-jump without a second's thought.
GILLIAN: What are you talking about? I'm coming with you.
KIRK: You can't. Our next stop is the 23rd Century.
GILLIAN: What do I care? I've got nobody but those whales...
While the clear implication is that she has no family and friends, there's nothing specific beyond that within the main canon. She doesn't appear in any future films or within any of the TV shows.
In the absence of anyone to push the police, a missing person report would probably be filed. Theoretically the file would then be closed several hundred years later when she re-appears.
EU Canon.
There are several mentions of her in authorised fiction.
She makes a brief return in TOS: Debt of Honor, set some 10 years later.
She attended Kirk's memorial in 2293
Many individuals had spoken throughout the morning. Dr. Gillian Taylor, a twentieth-century cetacean biologist who had helped the captain and the Enterprise’s command crew bring a pair of humpback whales into the present in order to avert Earth’s destruction, had recalled meeting Jim, being charmed by him, and at last being enlisted to his cause
The factbook Star Trek: Federation - The First 150 Years lists her 2299 book in its fictional bibliography
And the (chronologically) final mention of her is in 2310, as part of an expedition involving Captain Uhura, Carol Marcus, a mysteriously missing humpbacked whale and a renewed Genesis Project.
It wasn’t God who had summoned the darkness, but a woman named Carol Marcus. Even without God making an appearance, this was Genesis: the birth of life on a lifeless world. The rebirth of a failed project. But no birth is without danger. Lost in the birth-tempest was a young humpbacked whale, named Harpo.
In the observation room of the drift-station Madrigal, Gillian searched the waters and swore.