No, it did not.
Under Spain, viceroyalties had their own flags, and in this case the flag of New Spain was the relevant one. After independence, Mexican state flags and coats of arms were slow in developing. To compare to two more populous provinces, Jalisco only created its flag in 1972, and Durango still does not have one (it does have arms).
You are conflating the concepts of Nation and Country.
a large aggregate of people united by common descent, history, culture, or language, inhabiting a particular country or territory.
a ... territory considered as an organized political community under one government.
Nationalism - the concept that the populace of a country is best ...
It appears that that geographic isolation was a fairly effective barrier, and I find little mention of foreigners living in the New Mexico region.
A couple that are mentioned are noted in The Works of Hubert Howe Bancroft: History of Arizona and New Mexico. 1889
In 1804 William Morrison of Kaskaskia despatching the creole trader
Baptiste Lalande up ...