Kind of, there is a torque produced proportional to the applied load that counters the rotation, as clearly must be the case to cause mechanical power to flow from the prime mover to the machine rotor, but in a grid connected machine the speed is set by the grid frequency, not the load on the individual machine (Ok, there can be some fun dynamics involved, but assume steady state).
The dynamics are made complex by the fact that rotor speed (and thus frequency) is the integral wrt time of the net torque.
Consider an ideal three phase machine connected to a large grid, if we start off with everything in sync and no net power flow then the angle between the rotors field and the grid connected stator induced field is zero and no torque is generated. If we now open the throttles what happens? The prime mover tries to speed up, which causes the rotor to start to lead on the fields produced by the grid connected stator and as this angle starts to increase, a torque is produced as the magnetic field of the stator pulls on the rotor, eventually this torque equals the torque being produced by the prime mover and we end up in a (hopefully) steady state where the rotor is leading the grid by exactly enough to match the torque from the prime mover.
The interesting thing is that if the grid is massive compared to the generator, the speed has not changed (as that is set by the grid frequency), only the rotor angle has changed but we are now generating power.
If there is insufficient power input in total, the grid frequency falls (And all of the generators slow down), if there is excess power input in total the grid frequency rises, but in all cases the speed of all the machines matches the grid frequency.
Note that there has been no mention of voltage, which is usually controlled to set the reactive power, not the real power.