As I understand it the concept of what a quadrant is wasn't established until the beginning of TNG, in TOS it was used rather loosely as a means of describing a location. It was only in TUC that the TOS crew used the new format (Sulu and the Excelsior
were in the Alpha Quadrant when the Enterprise
-A was in the Beta).
Commander Austin Harris, First Officer, Deep Space Nine (by FltCpt. Bossco)
8.01 - Darkest Before Dawn
(Chapter 8 added, 12/09/2015)