Not at all, because the physical shutter is not used at all during recording video. It just stays open all the time.
Well...i guess one could count this as a single shutter actuation for every recorded video.
I suppose you had something in mind like 25-30 shutter actuations per second, racking up an insane shutter count very quickly, but that is NOT the case.
Edit: Forgot the NOT ;)
Edit 2:
What you set for video is frame rate, not shutter speed. This is very similar to shutter speed in the sense that it limits the duration of the exposure of an individual frame. But it does not use the actual physical shutter of the camera. Instead it controls the duration of the recording on the sensor electronically.
Keep in mind, the physical shutter is a mechanical part. It makes a noise when it is used. You can't usually hear it because the mirror slap noise is even louder, but it does have a distinct clacking sound that one can hear e.g. when takes a photo with mirror lockup.
Since you don't hear a machine gun like clacking sound during recording video, you can rest assured that your camera's physical shutter is not used at all.