1. Is it legal to shoot fox in your backyard where you live? If you have neighbors that are close enough you are worried about disturbing them with the already quiet .357 out of a rifle, then I cannot help but suspect you are asking for help in doing something that is illegal.
2. Are silencers allowed by the State you live in? Many States prohibit them.
3. Do you have any felony convictions? That will prohibit you from buying a suppressor. If not are you prepared to pay for the silencer, then fill out the paperwork, including a signature by your local Chief Law Enforcement Officer and two fingerprint cards, and then mail that off with check for $200 to the BATFE and wait a couple of months for them to send you back your tax stamp so you can go pick up the silencer? Do you live in a state that allows silencers? You'll also have to pay a gunsmith to thread your barrel. You'll be into this operation for about $1000 by the time you are done.
It would have to be quite a fox problem to make this a good path to take in solving it.
Most states allow trapping of fox. That's quite and you can dispatch them with a club or a .22
States that allow suppressors: AL, AR, AK, AZ, CO, CT, FL, GA, ID, IN, KY, LA, ME, MD,MS, MT, ND, NE, NV, NH, NM, NC, OH, OK, OR, PA, SC, SD, TN, TX, UT, VA, WA,WV, WI, and WY.