Quiet in air pistols is a function of type (springer, single stroke pneumatic, etc.), power, and pellet. I don't have a broad base of experience with a lot of different air pistol. I shoot bullseye and my FAS 6004 is very quiet with negligible recoil.
Log on to Pyramid Air, Airgun Depot, or one of the others. They have a chat feature and I'm sure they can recommend a quiet gun that might meet your needs.
Most target pistols are pretty quiet because they only shoot 350-500fps. I have a Crosman 1701P that I shoot inside, no hearing protection necessary. If you need enough power to subdue small pests, they also sell the 1720T (.177) and Marauder pistol (.22) that have shrouded barrels and shoot 700-800fps, more or less. All pistols mentioned are PCPs.
Get an olympic style 10 meter pistol. They are super accurate and relatively silent.
I do shot a IZH 46, and the hit of the pellet on the target makes more noise than the pistol.
Olympic type pistols can be very expensive, except a couple like the IZH and the TOZ(??).
This type of pistol will outshoot the human component.
I was failing to mention that target pistols made by Crosman, though look very crude, are excellent shooters. I have experience with the 1377 and the 2240, and I would recommend them for somebody in a tight budget. They won't disappoint you.
It's going to depend on what you'll be using it for and how much you want to spend. I have a Marauder Pistol in .22 cal. Out of the box it's borderline backyard friendly. Accuracy is ok once you find the pellet it likes. There's also a lot of aftermarket parts that you can upgrade as well. Mine has a Lane regulator, Marmot Militia Hammer Forged barrel, and added a LDC. It shoots alike a laser and mouse fart quite now. I've taken small game out to 60 yards. Also uses a magazine so follow up shots are easy.
I have a Weirauch HW 75 that is pretty quiet. Also my favorite to shoot. Very much like the feel of a 1911. Not the cheapest but you sort of get what you pay for. German made too.. The Daisy 777, 747, 717 were also really quiet and pretty cheap. I have a 777 and it's accuracy always has surprised me. Might be quieter than the HW 75. Of course, quiet mean less power.
Looking for the quietest air pistol. Who makes it? Model?
The Benjamin Marauder Pistol, aka PRod, is OK on noise out of the box. There are several sound moderators available that quiet it to zero. My choice is a replacement baffled barrel shroud from "Rocker1" on the Gateway to Airguns forum. It adds only four inches to the OAL and there is absolutely no muzzle report, even though I have increased the power on mine significantly. The only noise is the action and hammer fall and the thunk of the pellet hitting downrange. TKO makes an effective moderator unit that attaches to the end of the barrel, but it is longer than the added length of Rocker1's shroud, plus it costs more. If you find the lowest price for the PRod on the 'net and get a Rocker1 shroud, you should put out less than $450.
The peak noise from an air-gun is mostly determined by power level and power-plant type. Some power-plants just make more noise than others. All else being equal the following is loudest to quietest.
PCP (pre-charged pneumatic, these are the guns that require a separate pump or compressor and/or SCBA bottles)
MSP (multi-stroke pneumatic, like the old Sheridans and Benjamins that you had to pump 8 or 10 times between shots)
CO2 (the little 8 or 12 gram bottles that get pierced when you screw the cap back on the gun)
SSP (single stroke pneumatic, match guns and some Daisies)
Obviously, if you add some type of moderating device to the muzzle, you can further reduce the noise from any of the above. The noise from a PCP can be reduced by reducing the power setting. MSPs can just be pumped fewer times. CO2 guns can be tuned down by lightening the hammer/spring to reduce valve opening.