We are talking about cartographical conviences used to express varied political relationships, not real galactic structures, no? A quadrant is just a section of a larger territory that has been divided in four; any such division could be described as a quadrant. Moreover, the quadrants could be redefined in order to be useful to the context under discussion.
Could the four-part division of the Milky Way have been reconsidered at some point? Why not? Before WWI, the West consisted of nations and territories west of the Rhine River. During the Cold War, the West stopped where the Soviet Bloc nations began. Now, the notion of Central Europe is gradually replacing the East/West divide, with nations like Poland and Czech Republic being considered geo-politically alongside Germany and Austria. We have broad categories, like the Middle East, which at its largest consists of territories that are more west than Western Europe (Morocco).
Perhaps the alliance with the Klingons forced a reconsideration of the greater geo-political structure of the area round the Federation. Rather than Earth dividing Alpha from Beta, political planners decided that Alpha was a better description for an area under the influence of the Federation, Klingon Empire, and Romulan Empire. There's no need for these cartographical conveniences to be set in stone.