Main Canon
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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**...*

> <sub><sup>[Star Trek: The Voyage Home - Original Screenplay][1]</sub></sup>

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.


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EU Canon.
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There are several mentions of her in authorised fiction. 

**She makes a brief return in [TOS: Debt of Honor][2], set some 10 years later.**    
![enter image description here][1]

  [1]: https://i.sstatic.net/YU2iC.jpg
  [2]: http://memory-alpha.wikia.com/wiki/Debt_of_Honor