Not unless you are granted special permission. It is unlikely that the permission would be granted and it is also unlikely that you qualify for housing based on the non-marital relationship.
First, military bases are secure places. We do not like people who we do not know and trust to be living in our quarters. ederal law describes the various housing allowances and why and when you get to move into certain housing.
Your significant other (and you) would probably be in violation of a number of base policies and laws in the event the housing was for an unmarried couple on base. You could not use that person to increase your off base housing allowance and unless things have changed dramatically since 1997 when I was a Judge Advocate in the Marine Corps you probably cannot use that relationship to live off base.