The Milky Way galaxy's a big place. So much so, that 99% of Trek takes place in just one sector of it: the Terran-centrically designated "Alpha Quadrant". Because I'm sure The Dominion, for example, refers to the Gamma Quadrant as "The Alpha Quadrant"! lol So much so, that 2 other quadrants, Delta & Gamma, are thought of the outlands. Areas that are so far away as to be considered the middle of nowhere. Indeed, the only reason the factions of the those 3 quadrants even know about each other at all is because of traversal anomalies: Q zapping the Borg from thier Delta to the Federation's Alpha and a wormhole connecting The Dominion from thier Gamma to the Federation's Alpha. If not for those 2 things (Q & the wormhole) all 3 factions (Federation, the Borg & The Dominion) would have gone on thier respective merry ways for who knows how long; blissfully unawares of the other's existance all because of the immense size of a Milky Way quadrant. Well this, of course, apply to the last of the four of the Milky Way quadrants: the Beta Quadrant. There could be there a Federation equivalent: a benign organization of worlds all committed to exploration & mutual betterment. Or, of course, there could also be a Romulan Star Empire, Cardassian Union, Sheliak Corporate, Borg & Dominion equivalent's, too. All, because they never experienced a "Q-like Event" or a "Wormhole-like Event", are also going about thier respective merry ways for who knows how long; blissfully unawares of the Federation's, the RSE's, the CU's, the SC's, the Borg's & The Dominion's existence. Again, all because of the immense size of a Milky Way quadrant. So .... How about some sort of interaction in, if you'll pardon the unavoidable and wholly unintentional pun, that space. The Beta auadrant would be the quintessential Terra Incognita. A huge, fresh place where everything would be new to a Starfleet crew. A VOY in reverse, if you will. But this time, it's a ship & crew who know full well upfront that it's a one-way trip, but, in keeping with the spirit of the Fed's charter, are willing to make the sacrifice for the sake of exploration: be it of planets or of people. And seeing as how every Trek spinoff, except for TNG, is named after the structure that's the main setting of the show (be it a starship or a spacestation), you could name this Trek series after some historical ship that had the same mission statement: that is, being a voluntarily made one way trip to explore & make others aware of the existance of you & yours. Just an idea.
I find it a bit hard to believe Starfleet would purposefully send a ship out, never to return--it would be a lot to ask a crew, even more so if the ship included families onboard. There would be long-term missions, taking years to complete, but ships need maintenance at starbases, refuel and take on supplies--not all of which is guaranteed if left to find it from alien outposts. The UFP does extend into the Beta Quadrant, as far as the Klingon and Romulan Empires allow, but there will be large sections left uncharted by Starfleet ships, so there is a great deal of scope for stories.
"Q zapping the Borg from thier Delta to the Federation's Alpha " Nope, Q zapped the E-D 7,000 light years away, still well in the alpha (or possibly beta) quadrants. There's evidence that the borg were hovering around the Romulan neutral zone before this incident though. The Klingon + Romulan empires are generally accepted to be in the beta quadrant, although DS9 played fast and loose with distances, going from Frontier Medicine to about 50ly from Earth. The idea of long term exploration was the premise of Enterprise. It didn't last long. Voyager was the best you'd get with exploration, because the writers couldn't take them back to Earth when ratings started to slip. In the books Titan is off on a long term exploration mission (of the Beta quadrant as it happens). Events still conspire to bring them back to the fold on more than one occasion though.
Yeah, yeah, yeah. You're right, of course. I slightly glitched that one. C'mon, look at the time I wrote that: I was kinda loopy! lol Regarding the other inhabitants of the βeta Quadrant, i.e., the Klingon Empire, The Romulan Star Empire & the Gorn, plus the fact that the Titan's current raison d'etré is mapping that very area, it seems I was a tad off when saying that the βQ was a sector just ripe for story mining. Apologies.
I never really cared for the whole splitting the galaxy into fourths thing, but I guess they wanted to be able to quickly convey how far the Barzan wormhole extended.
The idea of the line between the Alpha and Beta quadrants running through (or near) Earth came up with the line "Enterprise is the only ship in the quadrant", but of course there are two meanings of quadrants -- the 1/4 of the galaxy meaning, and the much smaller meaning (there's at least 904 quadrants in TOS time) Excesior was mapping the Beta Quadrant in 2293, in TUC, which was the first use of the 4-quadrant system in the 23rd century (although not in production chronology - where The Price was the first mention, nor in trek chronology, as the 4 quadrant system was present in Enterprise). As Praxis exploded and affected Excelsior, but we didn't see evidence of other ships being affected, there's a good chance that Praxis was in the Beta Quadrant. However the OP's late night ramble remains true, plenty of space left in the beta quadrant (25,000 million stars). Plenty of stuff left in all the quadrants though.
In every depiction I've seen of the galaxy in the Star Trek universe -- granted most of them non-Canon -- the Federation and its neighbors have always taken up not much more than a miniscule amount of space surrounding the Alpha/Beta boundary where Earth sits -- when compared with the size of a quadrant. There are likely years of exploration left in the mid to far reaches of the Alpha and Beta quadrants and maybe a couple of civilizations the size of the UFP, RSE, etc.
Here's a (very large) map of the alpha and beta quadrants. Thought I should add it to this thread. http://www.chartgeek.com/wp-content/uploads/2012/09/star-trek-map.jpeg