The Overlord wrote:
I think a more likely explanation is Quadrants meant something much smaller in TOS, but where retconned into being a quarter of the galaxy in the TNG era, so the talk in TOS of the Klingon and Romulan Empires being in the Beta Quadrant doesn't mean anything in the TNG era cannon.
That is the way I see it. The explored galaxy was much smaller in TOS. As exploration expanded the known galaxy the terms had to expand as well.