Can an aircraft take off with zero ground speed?
The speed in relation to the ground is completely irrelevant. The only thing that matters is the speed relative to he air. (It's called an "airplane", after all.)
So, if you can get enough air flowing over the wings without moving the airplane relative to the ground, then it will take off. The takeoff speed of an F-22 Raptor, for example, is somewhere around 130 kt ≅ 150 mi∕h ≅ 240 km∕h, so the required wind speed would be equivalent to a category EF3 tornado.
If you can generate that much airflow, then yes, the jet will take off.
If there is enough headwind, yes, as that's the only thing that matters, groundspeed is of no relevance.
If you're asking about the treadmill, see here.