Eurasion collared doves, yes, but not Mourning doves.
From the New Mexico Department of Game and Fish 2016-2017 Hunting Migratory Game Bird Supplement publication:
Legal Sporting Arms
Migratory game birds may be taken with a shotgun fired from the shoulder, provided it is no larger than 10-gauge and incapable of holding more than three shells, or it is plugged with a one-piece filler that cannot be removed without disassembly which renders it incapable of holding more than three shells. Migratory game birds also may be taken with muzzle-loading shotguns firing shot, bow and arrows and crossbow and bolts.
In addition to the New Mexico regulations set forth in this booklet, the following federal regulations apply to taking, possessing, transporting and storing migratory game birds (ducks, geese, coots, common moorhens, snipes, soras, rails, doves, band-tailed pigeons and sandhill cranes). Persons cited for violating federal regulations will be required to appear in federal court. The complete federal regulations (50 CFR-Part 20) is available online:
https://www.fws.gov/le/pdf/50_CFR_20.pdf.
In that federal publication, it states that migratory birds may not be taken with a rifle or pistol.
Pellet guns are not mentioned as a legal method for taking furbearers including racoon, coyote, skunk...etc.
Looks like the only game you can take there with a pellet gun are grouse, Eurasian collared-dove, and squirrel.