And Pavonis, I agree -- the Federation and its neighbors are called "Alpha Quadrant" powers even though many are in the Beta Quadrant, in the same way Europe is considered part of the Western world even though most of it is in the Eastern Hemisphere.
The reason that Europe is thought of as being in the West is because it coincides with the Western Roman Empire, much more so than with the Eastern Roman Empire. Our designation of Eastern versus Western as applied to civilization was originally established relative to Rome. On the other hand, the hemispheres of the globe were designated relative to Greenwich, England, a designation which came much, much later.
The situation with the Alpha Quadrant, Beta Quadrant, and the Federation cannot be the same as this unless there is an older convention which at one point placed the whole Federation more entirely in the (older conception of the) Alpha Quadrant.