The Four Corners region or the specific monument?
A little tricky for me to check right now, because I gave all my reference material to my daughter, who is on her way out to SW Colorado.
In the broader Four Corners region, I would stay at each place I visited: maybe Gould's for Monument Valley, any of the commercial parks at Moab (I've used the KOA), and I've stayed in Farmington on the way out of the area. Durango, I don't know, haven't been there yet. For the Colorado National Monument, I stayed at a KOA in Montrose, and a former KOA in Green River, visiting the monument enroute that trip.
An earlier trip to that area, studying geology in the region, I stayed a week at a hotel in Green River, because it was closer than Grand Junction to the outcrops we needed to visit.
The Four Corners monument itself, there is nothing really nearby, as there is not much commercial development on the Navajo and Hopi reservations, and the reservation boundaries are extensive. In that area, however, the concept of "nearby" means not much more than a two-hour drive.
There is a "working man's" RV park in Kirtland, just outside the reservation on US-64. that might be where I stayed, rather than actually in Farmington. If you are coming from or going to the north, on US-160, the nearest developed area off reservation would be Cortez.
You will find campgrounds at some of the national monuments in the area. It has been a while since I've been through, but when I was last there the facilities were rather limited. Campgrounds, not RV parks.