Is it quiet in the cockpit when flying faster than the speed of sound?
No, for two reasons:


Speed of sound is always relative to the material the sound propagates in; it moves faster through denser media, such as the aluminum structure of the aircraft itself. The aircraft's structure moves with the aircraft, so the vibrations of the jet engines will be transmitted to the cockpit/cabin at the same speed as when the airplane were flying subsonic.
The air inside the cockpit is still, and so sound within this local bubble is unaffected by the speed of the bubble itself (and so sound can travel faster than sound). If this weren't true then radios and other cockpit sound sources like alarms would be inaudible at supersonic speeds, which would be a major safety hazard and tactical disadvantage to supersonic flight.


You actually don't even have any significant aural cues inside the cockpit that you've exceeded Mach 1; you have to rely on your airspeed gauge to tell you.