In TOS, I think a quadrant was supposed to be just some marked-off region on a star map, and the Federation contained many quadrants. TNG-era terminology re-defined a Quadrant, capitalized, as one-quarter of the whole galaxy.
I suspect that a lot of Star Trek fans, and some of the writers, had no understanding of how big our galaxy is.
- Our galaxy appears to contain up to 400 billion stars. That's a lot.
- Even if you could travel at 1000 times the speed of light, it would still take decades to get from the heart of one TNG-era Quadrant to the heart of another [or to get from Earth to the center of the galaxy, as they did in ST-5; the spinoff TV series had much better science advisors than Shatner did for his film].
So the galaxy is an absurdly large territory and far bigger than the whole Star Trek franchise would ever need to tell all its stories, as long as its size is understood.
The Borg were placed in the Delta Quadrant and the Dominion in the Gamma Quadrant, both vastly far away, in order to explain how we had been isolated from them for so long. Things like Q's magic, trans-warp drive, or a wormhole are needed to cross the distance.
As for the Klingons and Romulans, the only possible way they're in the Beta Quadrant is if the Federation is right near the Beta border, and these two alien empires are right near the Alpha border, like R. Star said.