No, the electric motor only turns when it needs to. Either to move the car, or to start the ICE (you'll see that acronym a lot on here. It means "Internal Combustion Engine", the gas-burning part.)
The way the Prius' transmission works, the ICE, one of the two electric motors, and the wheels are pretty much always connected. So if you spin one, at least one of the other two has to be spinning, too. If the car is stopped, then either both the ICE *AND* the electric motor are stopped, or both are running. So when you first turn on the Prius, after a few seconds, it needs to start the ICE warm-up cycle. It spins the electric motor to spin the ICE (same basic idea as a conventional car's starter motor,) but once the ICE is spinning on its own, the ICE spins the electric motor, which generates electricity to recharge the battery. So it's a 'two-for-one', in that warming up the ICE also recharges the battery.
Once the ICE is warmed up, (assuming you haven't started moving yet,) it shuts down, which also stops the electric motor, so your battery doesn't get charged any more.
Once you start moving, the car decides how to apply power. Either from the electric motor only, leaving the ICE stationary (again, starting one of the three items turning means that at least one more has to, rather obvious when trying to apply power.) Or we can run off both ICE and electric, adding together their torques to drive the wheels. Finally, when we're up to a reasonable speed, it may choose to use only the ICE. When only the ICE is powering the car, we have one of two situations: either all of the ICE's power is used to turn the wheels, in which case the electric motor is stationary; or the ICE may be actually producing more power than is needed to drive the wheels, in which case it also spins the electric motor, charging the battery.
As long winded as this is, it's also an over-simplification, since there is a second electric motor in there that operates differently from the one I was talking about above. (For example, it may have the ICE spin one motor, creating electricity, that then gets sent to the second electric motor to help drive the car. It may seem silly, but it's great for converting horsepower into torque more efficiently than the gas engine alone could accomplish.)