You will recieve this quest from a spouse when you visit them as they will give you a letter. It turns out its from a man named Kenneth in Bowerstone who claims to know you are an adulterer and wants money to keep your secret.
There are a few options to completing this quest. You can travel to Bowerstone and pay the man - he only wants 2,000 gold pieces, you can ignore the letter and let him tell your spouses about your extra-wedlock shenanigans or you can kill him.
No matter how it unfolds, when one of the three options happens, the side-quest will be completed.