I don't have a scientific answer to your question but I think that Star Trek in all of its varied formats has all taken place within the Milky Way. The ill-fated crew of Voyager were cast adrift in the delta quadrant - but even that was within the Milky Way. There were some plot lines involving alients from the Andromeda Galaxy - but they travelled to the Milky Way rather than Federation vessels travelling there.

When you consider that the Milky Way has between 200 and 400 billion stars and is a disk which is around 100,000 light years across then there are plenty of place to bodly go without leaving our galaxy.

Writing casually from memory about Star Trek is dangerous so I await correction from super-fans of the series. I would add the caveat that I can't recall where the Q came from...and also that I probably need to get out a bit more.

It is practically dead certain that there are vast numbers of stars between galaxies, just very thinly spread, and probably not very bright. We therefore would find it very difficult to see any of them, let alone be sure which were on the way out.

Bright stars burn out quickly, leaving only dwarfs and gas clouds, and in any case, there are far more dwarfs than bright giants.  It follows that, by the time that a star gets a few light years outside the nearest galaxy it probably would be a dwarf or neutron star whether it started bright or not. 

Anyway, what counts as "outside a galaxy"? I reckon a star would need to be hundreds or thousands of light years away from the nearest stars inside the galaxy to count as "outside". That might sound very far, but remember that our nearest galactic neighbours are of the order of hundreds of thousands or millions of light years away, not mere thousands of light years.

A typical sizeable galaxy is tens of thousands of light years in diameter and very ragged in appearance, so to define its boundary is quite a problem, given that a typical star would be travelling at only a few tens to hundreds of km/s, so it would take a long time to travel so many light years. Assume 100 km/s, quite a high speed for a star near the outskirts of a galaxy, and it would take something like  a million years to travel about three hundred light years, barely out from mother's apron. To get halfway between us and Andromeda, our nearest spiral galaxy among our neighbours, would take something like the age of the Earth. roughly 4 billion years.  Forget the dinosaurs, during that time we evolved, not just from pond scum, but from before the first functional cells!  And a star a bit more than twice the mass of our sun would be a dwarf after just 1 billion years, so you can see that we would have a pretty hard time of it to see anything between galaxies.

At the same time, when you are dealing with hundreds of billions of stars in mutual orbit, which is what a galaxy amounts to, you can be sure that some of them would be "slingshotted" out into escape trajectories.  You could say that those stars "evaporate" from the galaxy.

A more drastic boiling off would result when two galaxies collide, as galaxies do from time to time. Probably within a few billion years Andromeda and the milky way will be scattering each other's stars into outer space by the million. Talk about letting off steam!

Just vapouring...

Jon