if vibration is from engine, chances are sound will be there regardless of gear position.
from your description, there's no sound when gearbox is disengaged (i.e. at N, P) but sound appears when gearbox is engaged (i.e. at D). vibrations may be caused by the engaged clutch when car is at "forced" stationary; notice that car moves forward when you release the brakes at idling (level ground). these vibrations might be the culprit. but why no sound at R, no clue liow.
btw, my mech advised me use N gear whenever i stop, suggesting that this may preserve the clutch and also the brakes which otherwise need to be working to keep car stationary... how true i'll probably check with TCM during my next servicing.