I have ony lived in New Mexico for about a month but I want to get divorced, can I file for divorce here?

New Mexico has the same domicile requirement as California.  You must have lived and had a domicile in New Mexico for 6 months before the state has "jurisdiction" over your case. 

Jurisdictional time requirements are controlled by statute and came about because people would fly into another state with lenient divorce laws, get divorced, then fly home.  I believe you can still file for separation in the meantime without the time requirement.  Check with the court clerk.