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...
Star Trek: The Voyage Home - Original Screenplay
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. The final mention of her is that she's been given her own ship and sent off on a mission to recruit some divers to work with her whales.
"You're going to your ship, I'm going to mine. Science vessel, bound
for Mer to recruit some divers to help the whales. Why, the next time
you see me, I may have learned to breathe underwater!" She grinned,
honestly and completely happy for the first time since Jim had met
her. "I've got three hundred years of catchup learning to do," she
Star Trek IV: The Voyage Home - Official Novelisation
For the record, the factbook "Who's Who in Star Trek 2" identifies her ship as the USS Clarke and her organisation as the 'New Cetacean Institute'.
There are several mentions of her in authorised fiction, both relating to the present day and her future in the 23rd century.
In the story "Whales Weep Not" we learn that her disappearance was reported to the police by her landlady. The novel also confirms that she has no family or significant ties.
“She doesn’t have any family. Only child, you know. Her mother died a
couple of years ago. Cancer if I recall.” She scratched at a curler
and thought for a minute before adding, “she never talks about her
dad. I figure he’s long gone. Only steady boyfriend she had quit
coming around when her mom got sick."
and that the police left her case open, in the event that she turned up again.
The case couldn’t officially be closed but that didn’t mean he was going to put any more time into it either. After placating Mrs. Schimmerman with some story he would quietly add the folder to the inactive file. He just hoped that Dr. Gillian Taylor was happy with her whales.
She makes a brief return in TOS: Debt of Honor, set some 10 years after her arrival in the future.
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
Crucible: Spock - The Fire and the Rose
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.
Strange New Worlds II - The Hero of My Own Life