Legal Question in Traffic Law in Georgia

Yes. They can:

(1) Issue a warrant for your arrest and extradite you to New Mexico (although unlikely).

(2) Ask Georgia to suspend your license until you pay.

(3) Hire a collector and pursue you here in Georgia (and ruin your credit).

Why haven't you paid it instead of risking these things?

They can, and there are several indirect methods for them to do so. The question is going to be what sort of effort they make. The easiest for them is to just put a warrant out on you for failing to appear in court for the ticket. If you are ever pulled over, there is a good chance that the warrant will show up, causing you to be arrested. You will then sit in jail until New Mexico decides whether or not they want to come and get you and bring you back. If they don't, you will have sat in jail in the meantime, and face the possibility of this happening over and over. If they do, there will be trouble when you get back there.