What is the quietest LEGO motor?
I don't think it would be practical to evaluate every LEGO motors sound output, almost all LEGO motors are fairly noisy so I think it is best to address the problem of a noisy motor in general.

My first suggestion would be to remove the source of the noise first, by using a purely manually operated dolly, if this isn't an option (perhaps you would prefer to have the precise motion given by motors) then I would consider moving the motor as far as way as possible from the microphone, perhaps using axles to distance the power drive (similar to the long steering axle used in the video I linked above).

The faster you run a motor, the louder they will be, so it might be possible to run the motor at a fairly slow speed, using a high to low gear ratio from input to output you can increase the speed without loud motors. However the sound of too many moving parts (gears, axles, etc...) may contribute to the noise.

As others have suggested in the comments, could you use a separate microphone instead, positioned away from the motor? Think about how you could possibly insulate the motor from producing too much noise using sound absorbing material.

Consider using video or audio editing software that can remove the noise. Adobe Audition for example has an amazing Spectral Display feature that allows you to visually identify certain frequencies and remove them, the results will vary based on the quality of the audio. 



Motor noise will always be a problem, consider 'out of the box thinking'. In summary:


If you can build something without a motor, do.
Move the noise away from the microphone
Reduce the speed of the motor (adjust speed with appropriate
gearing)
Use a separate microphone (or dub audio)
Use software to remove motor noise