Each base is different. The policy is set by the base commander. Here is the policy where I have been stationed in base housing with weapons:
1. Permission must be obtained from the individual command Commanding Officer (CO). The form lists the resident's name, address in base housing, and the firearms by make, model and serial number.
2. That form than goes to base security, where they keep a copy for firearms registration and they sign off on it.
3. The form then, that has been to the individual command CO and then base security goes to base housing office where they keep a copy of it.
4. There are more stringent rules for the actual storage of the firearm in the base housing unit: it has to be under two locks - one the base housing locks and second a locked closet or safe.
5. Finally transport on and off base the gun must be clearly visible and unloaded until off base.
This was the policy at two navy bases I have been stationed at. Ft. Hood, especially being Army, might be different. Is base housing even on base? If you don't have to go through a gate to get into base housing, I would say, Don't Ask, Don't Tell. Inform the base housing resident first as they have a lot to lose if something goes bad. I'll try to do some research on Ft Hood rules and regs and post them on Monday.