The real problem is that choosing to divide the galaxy into 4 quadrants is not just arbitrary, it's also silly in that it doesn't follow any "natural contours" of the Milky Way.
It's the equivalent of drawing a line through Greenwich, England, and declaring that everything within 180 degrees to the west that's north of the equator constitutes the Earth's Alpha Continent while everything within 180 degrees to the east that's south of the equator constitutes the Delta Continent, regardless of the placement of land masses like Eurasia, North America, or Africa.
The real question the Star Trek writers should have considered -- and didn't most of the series have a technical adviser to help them? -- is whether the Federation and its neighboring powers (Romulans, Klingons, Ferengi, Cardassians, First Federation) are all in the Orion Spur, or whether explored space includes parts of the Sagittarius or Perseus Arms.
Their territories could easily fit into any odd shapes as needed within the 3-dimensional Orion Spur, allowing the writers to be nebulous, so to speak, about who borders whom.