How do I begin the Elderslayer quest?

It says to explore the maps in the corner regions of the atlas to find them. So does this mean I have to explore Lex Ejoris, Haewark Hamlet, New Vastir, and Lira Arthain zone maps until the Elderslayers randomly spawn? Plus I have a watchstone in one of the regiions (I only earned one so far), does this affect the spawn rates of the Elderslayers?

Yes, you should spawn one (different) conqueror in each of those regions before you try to spawn them in the inner regions.

The stones change the tiers of map showing in the region, so they do influence the spawn mechanics, as well as what maps can drop. e.g. once you have a stone of each color from those corner regions, you will need one stone socketed in the citadel of a region to spawn your next conqueror of any color.

There are quite a few good videos on atlas mechanic details. Best to watch one from the last couple of months so that you know it's current.