RE: Quietest deer gun/caliber
The problem you may run into is barrel length. A 44mag has a low volume of powder, but is generally offered in carbine lever actions. The longer the barrel the less noise, So you need a minimum of 24" in a lower caliber. The 6.8 spc comes to mind. if you don't reload. If you do there are countless cartridges that will fit the bill.
Blessings