Gamma Quadrant is a term invented and used by the sentient beings of the Alpha Quadrant - yet, how do the "natives" of the Gamma Quadrant call the area of space they inhabit?
There is no canonical answer to this question. We always hear Dominion characters translated, with their cartographical references translated as well. Unfortunately, the writers never provide even a throw-away line suggesting how Dominion astrocartographers (or, for that matter, Klingon or Romulan ones) might view the galaxy.
In the Deep Space 9 episode "One Little Ship", there is tension between the old breed of Jem'Hadar and the new breed. The distinction between the breeds/generations is where they were born. Those born on the other side of the wormhole are referred to as Gammas, those born on the Federation side are referred to as Alphas. At least in this situation, they seem to have adopted the same terminology the races of the alpha quadrant use.
Why they do this is unexplained -- they may have picked up the terminology from alpha quadrant races, or it may be how the universal translator decides to translate whatever native terminology they have.
This whole thing reminds me of a great xkcd strip:
The word "West" speaks of a direction relative to one's own location, but in many parts of the world the phrase "the West" has come to be synonymous with a loose collection of nations in Europe and North America; a political entity, more than a geographical one.
So, I suspect, is the case with this talk of Alpha Quadrant vs. Gamma Quadrant: The Alpha Quadrant powers are actually a loose collection of political entities from various parts of the Alpha and Beta Quadrants ... and at least a few Alpha Quadrant governments actually decided to side with the Dominion.
It's quite possible that because their first and foremost contact was with people who referred to this side of the wormhole as "the Alpha Quadrant", the name - or some aspect of it - stuck. However, as riffraff mentions, we don't actually know how these alien species refer to the Alpha Quadrant powers in their own languages; all we know is that when they speak English, or are heard in English through a universal translator, they appear to say "Alpha Quadrant".