Tim, the quadrants are divided up from galactic center point not the earth. It's unclear if the system predates the federation it might well be based on an older Vulcan system (though renamed) like the Class M designation

The Klingon Empire and the Romulan Star Empire are generally considered to lie in the Beta Quadrant.

Most federation holdings are in the Alpha quadrant but Sol itself is generally denoted as being at the border of the alpha and Beta Quadrants

The four quadrants of the milky way have been shown many times on screen and divide the flat plane of the milky way.

Earth by the way lies on the border between the Alpha and Beta Quadrants