TOS used the terms "quadrant" and "sector" interchangeably
...But ST2:TWoK specified that a quadrant was a subsection of a sector. Which can be retroactively applied to all the TOS references without a problem. It can also be applied to early TNG references to quadrants, such as the Morgana Quadrant: there simply are two types of quadrant, the local and the galactic, and TOS never concerned itself with the latter because there were no wormholes and the nearest thing to Q only ever flung Kirk's ship a thousand lightyears.
We could still have it be a case that the Klingons are mostly in the Beta Quadrant but have some territory or colonies in the Alpha Quadrant as well.
And this would be enough to set up the situation where, late in DS9, Klingons are considered an "Alpha Quadrant species".
That is, if Klingons were mostly in Beta, they'd originally certainly be known as a Beta species, the way Russkies were the Eastern foe. Indeed the whole Alpha/Beta division might have been devised to separate "us" from "them" in a putatively astronomical but essentially political way. But once they became solid allies in the war against the Dominion, they'd be "upgraded" into an Alpha culture, the double excuse being that they operate in Alpha for shared Alpha goals, and a corner of their territory also extends to Alpha. Basically, Russia counts as the West in certain contexts already...