This question is at the best, ambiguous. There can be both yes and no answers based on what is being done with the aircraft and treadmill. The point is that for an airplane to lift off, there should be sufficient airspeed. If there is no wind, there airspeed is equal to the ground speed
Assuming that there is no wind (into or against the aircraft), there are two possible solutions.
If the airplane is stationary relative to the ground, it won't take off (as wind speed is zero).
If the airplane moves relative to the ground (with sufficient speed), it will take off.
Assume that we have a jet airplane (just for sake of argument) and some one pushes the throttle and it begins to move forward. Now, as the treadmill has an infinitely adjustable speed, we can have three conditions:
If the treadmill speed is zero, the airplane will eventually generate sufficient lift and take off.
If the treadmill speed is adjusted such that the airplane is kept stationary relative to the treadmill, the airplane will take off (as it is moving with respect to ground, and so has some airspeed).
If the treadmill speed is adjusted so that the airplane is kept stationary relative to the ground, the airplane cannot takeoff, as the ground and air speeds are both zero. Note that in this case, the aircraft speed relative to the treadmill is twice that of the speed at which the treadmill is being operated.
If there is a wind, the required ground speed can be adjusted accordingly, but the principle remains the same. For example, if the wind speed is equal to the airspeed required for take off, the airplane will lift off even though it is stationary with respect to the ground.
Again, the important concept here is airspeed. It does not matter if the aircraft is on a treadmill, train track or runway.