It is not your Virginia non-resident permit that makes OC in NM legal however - it is the legality (no law against it) of OC by all non-restricted persons.
New Mexico is one of our "Gold Star" open carry states. There is complete state preemption of all firearms laws, open carry is increasingly common and law enforcement is well educated as to its legality. In addition, New Mexico has an extended domain law which effectively makes your vehicle an extention of your home. So you may carry openly or concealed in your vehicle.
You might wish to confirm that non-resident permits are recognized - I find no information to the contrary though