After the Mexican Revolution, there was a province called Nuevo Mexico, that somewhat is in the location of current-day New Mexico. Did that province have its own provincial flag? If so, what did it look like?
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).
While today's New Mexican flag is based on a pre-Hispanic motif, during the Mexican period the only relevant flags were those of the Mexican nation. These went through five iterations before the loss of New Mexico; note the disappearance of the crown in 1823.