I will assume OP is referring to the top right quadrant as the first quadrant, and then wondering why this is the convention.
It has to do in part (I suspect) with the European tradition of reading from left to right. Putting this together with the association of "up = more", when you have two strictly positive coordinates, this "generates" the first quadrant in the way with which we westerners are familiar. But since it becomes necessary to generalize to negative numbers (and also to have a familiar system in which trig can work), we have the "first-through-fourth" quadrant convention.
After thinking about it for a moment, it occurred to me that the above must have been first articulated by Descartes in his treatise on geometry, and once I actually checked, it seems I have some vindication in this.
Also, OP, in order for nice trig things to work (and so on), we must treat of quadrants as "consecutive" things. a "1-2-3-4" arrangement, or verbal description, wouldn't make sense from a trig (and later) standpoint, regardless of chirality/directionality.
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