1. No, because there won't be air moving across the wings, and therefore no lift
2. However, a treadmill moving at the take-off velocity of an airplane will create such a movement of air due to friction between the treadmill and the air, (called the boundary layer) that the windspeed will be considerable anyway
3. If the treadmill moved a few times faster than the take-off speed and the airplane was modified so it could taxi its wheels at such a speed, then you could achieve a boundary layer speed that would be enough to achieve lift-off
4. If the treadmill is too short, the aircraft will lift-off and accelerate while its in the boundary layer of the treadmill, but once it exits it it will suddenly be at a lower effective airspeed and lose its lift: so the treadmill needs to be long enough to let the airplane accelerate to an adequate speed before it enters the still air. Also, it would need to use its jet engines not its wheels
5. Flow from boundary layer friction is turbulent and has high vorticity; thats the effect of applying a shear force to a mass of air. This is 'low quality flow' and won't give as much lift as moving through still air would. So the treadmill would need to be a LOT faster than the take-off speed.
But hypothetically, yes, its possible. Smile now.