Can I install low voltage lighting in louvered wall register?
Not for ducts

The rules about wiring within HVAC ductwork are extremely stringent -- this is not due to anything related to voltage (i.e. shock hazard), but the potential for wiring insulation to overheat and smoke folks out.  I'm not even sure if you can run fiber optic cables through HVAC -- even that may be verboten.

But for a louver between rooms, this should be OK

However, since this is a louver connecting two rooms directly (no duct involved), you can put a low voltage luminaire there.  Either way, the precautions are the same: I'd use a type AC or type MC cable without an insulating jacket here, either that or a metal conduit wiring method, with the luminaire mounted to a metal box and using integral LEDs (vs an incandescent or halogen light bulb).  This minimizes fire and smoke risks.