Yes you can--so long as you're not on an Indian Reservation. You may want to check out the relevant state laws for other prohibited places:
http://www.opencarry.org/?page_id=278
Incidentally, legal residents (non-citizens) were not able to get a NM CHL until recently (thanks to RougeAussie on here). Though, interestingly, NM had no law against legal resident non-citizens carrying openly. So I guess all the times I had CC'd with my Texas CHL through NM I had been technically breaking their law. Kinda silly really.