Not necessarily. Holding still is useful, but a modern submarine isn't dramatically louder when in motion than it is at rest.
Sound from turbulent flow over the surface of the hull is minor and doesn't travel very far (IIRC it's more of a problem for listening than being heard). More significant is the noise generated the formation and popping of bubbles on the surface of the propeller (cavitation) as it moves through the water. But this is a function of depth in addition to speed: to prevent cavitation you can just go deeper. Or more precisely: the deeper you go, the faster you can turn your propeller without cavitating.
Sound from turbulent flow over the surface of the hull is minor and doesn't travel very far (IIRC it's more of a problem for listening than being heard). More significant is the noise generated the formation and popping of bubbles on the surface of the propeller (cavitation) as it moves through the water. But this is a function of depth in addition to speed: to prevent cavitation you can just go deeper. Or more precisely: the deeper you go, the faster you can turn your propeller without cavitating.