So I dont quite remember how I came across the cartographer quest that I'm on which is the Emerald Stair one.  I didn't do the Dawnshore one and after finding what I needed to find for the Emerald Stair one, I have no clue where to go to turn it in.  Can I even do the first one in Dawnshore now?

There is an emporium, I think in Dawnshore - Paradis City - High Market, you could try there. I think that the emporium is where all cartographer quest after the initial one are, maybe, centred. Sanza, for me, was originally at Magrans Assay in Dawnshore [if memory serves] - maybe try there! Good luck.

This happened to me too. There's no map marker showing where to turn it in, but if you travel back to Dawnshore, then go to Sanza's Emporium in Paradis Hightown and talk to Sanza, you'll be able to turn in the quest.

As for how you received it, it was probably when you accepted 6 quests from the bounty board in Fior mes Iverno all in one go. Five of those were bounties, the 6th is this map's breadcrumb for its missing cartographer.