Texas, primarily the western regions.
However it's origins are slightly obscure, the name is a diminutive form of the spanish Faja and translates roughly into 'Little Belt', as the original recipe calls for beef skirt and as such it is hard to define exactly when the dish was named Fajita in official terms.
Homero Recio conducted the first historical study into the Fajita in 1984, and found evidence of the dishes recipe dating back to the 1930's, although the word Fajita did not hit print until the 1960's.