There's no saving the many times early Trek used quadrant and sector interchangeably.
...Like we today use hour and minute interchangeably?
There's no problem with having both quadrant and sector. And the former seems to be a subset of the latter, at least in ST2:TWoK, where the presence of the
Reliant is increasingly unlikely when, as per Sulu's crescendo of statements, she's not just in the same sector, but also the same quadrant, and even showing signs of slowing.
Quadrants tend to occur when there's a specific destination there - perhaps a single star system, even. Which is fine, and the only real problem is that dividing a generic volume in four (rather than, say, eight) is much less convenient than dividing the flat disk of the galaxy in four.
KIRK: Captain Koloth. About that apology.
KOLOTH: Yes?
KIRK: You have six hours to get your ship out of Federation territory.
True enough. Since the story revolves around contested neutral territory, though, and since the Klingons apparently have a comparable "outpost" a mere parsec away, we're clearly talking borderlands here. Might be this qualifies - or then there might even be a special arrangement of neutrality as regards the outposts from which the two sides compete over Sherman's, requiring Starfleet to call its outpost "deep space station" rather than "starbase".
(Heck, perhaps the outposts themselves are territory, even when they don't reside within territory? Meaning that technically, Koloth could reverse his ship out of the parking lot and then remain loitering there, out of spite - but obviously he won't do that in this particular situation!)
Timo Saloniemi