22 with a can out of a bolt gun is pretty quiet.I shoot on sight at my house . They got in the attic in a house on my street and chewed up some wiring and burned the house down.
I also used catch and release traps you can buy from tractor supply. Glue down a pecan or some acorns on the trap trigger. Buy a tub big enough to but the trap in so you can drown them when caught.
Put a bird bath and bird feeder in your backyard with a good backstop. CCI 22 Quiet long rifle out of a bolt action rifle with a good scope with adjustable parallax.
CCI 22 Short HP is a nasty round if you plan to shoot up into the trees. Try to find a bolt action with a 24 inch barrel for this. Good glass or a peep sight is a must.