We are talking about cartographical conviences used to express varied political relationships, not real galactic structures, no?
Absolutely - and we may even conclude that these are conveniences specific to individual cultures. Klingons may not believe in Alpha and Beta Quadrants: it's just the Universal Translator giving us the impression that they do, much like it fools us into thinking they believe in "hours" and sometimes also "meters".
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.
If this happened between TNG "The Price" and VOY "False Profits", it explains a thing or two... Moving the A/B border would necessarily also move the G/D border, thus allowing for a wormhole end that in "The Price" was at the very most 200 ly into Delta to now be several thousand lightyears into Delta and thus on the route of the Voyager