I looked at the tables in the links given in that page and I saw that the number of non-Hispanic Whites was always lower than the number of Whites so truth is that most Hispanics of New Mexico identify as white.
From all I've read, the largest ethnic group in New Mexico are still Hispanos of New Mexico/Neomexicanos that are descended from the old Spanish settlers of New Mexico.
I mean in any case they have been assimilated into the English speaking community, without any 20th century Mexican migration I think the community would be pretty small.
Indeed, many Hispanos of New Mexico have assimilated into the Anglo community. Many of them no longer speak Spanish. However a very big ammount of people in New Mexico descends from the old Spanish settlers there. The famous Demi Lovato is one of them.
Also, looking better at the source of that Wikipedia link, https://www.census.gov/population/www/documentation/twps0076/NMtab.pdf, it can be seen that the Hispanic option was not available for a long time so many Hispanics identified as white. The term Hispanic as a race in censuses is somewhat recent. Hispanics are a plurality in New Mexico.
Do I need to? If non Hispanic Whites didn't became majority in New Mexico they surely almost become it, although I don't think the stuff you brought disproves that either
From the answer in the Quora link by Steve Tenbrink who lived in New Mexico for a long time, there wasn't a huge movement by Anglos to move to New Mexico as it was an unappealing and somewhat desertic territory. The Anglo population in New Mexico only really grew in the 20th century and it was never enough to turn New Mexico more Anglo than Hispanic. Hispanos are a plurality with Anglos being a little behind.
Also, the Hispanics of New Mexico can be split into two groups. One in the northern and central parts of the state is mostly descended from Spanish and Mexican settlers pre-cession. The other in the southern part of the state is mostly descended from more recent Mexican immigrants.
There was a Spanish governor of New Mexico in the 1770s who managed to military force a treaty on the Comanche. That explains the bigger Spanish settlement there than in the neighbouring territoies. That along with the smaller ammount of Anglo migration to New Mexico than to the rest of the Mexican cession explains why the Hispanos of New Mexico remained the largest ethic group.
If we look at older ages, we can see that between 85and 55 years, so about a generation, the population is majority White and it remains plurality up around the 45 years mark, so it seems like a generation and a bit more is White today, this could be due to different mortality rates but it still shows.