Luckily this doesn't happen too often out of the blue, but two recent times that I can recall:
While taking a bike ride with my older daughter, we happened upon an anti-abortion demonstration and we were approached by one of the demonstrators while waiting to cross a street. It was mostly a polite conversation but I made it clear immediately that we were both pro-choice and atheists.
A few days after my divorce was final I decided to take an Uber to a local bar just to get out of the house. The driver started with the small talk and being a bit buzzed already I told him that I recently finalized on a divorce and just wanted to get out of the house. Immediately he asked if I have asked God for help. I let out a chuckle and somewhat sarcastically said, "No. You're talking to wrong person about that because I don't believe in God."
Despite my efforts to drop strong hints that he wasn't going to change my mind, he kept on going on about it. The last few minutes of the ride I ended up going on a rant about people I know who have similar problems with depression and their marriages and how ultimately I took control of my own life and did what I had to do to make it better; and how these other people I know are miserable every day and every day they pray to God to help them or take away their pain and in the end not a damn thing is better for them. By this time we were at our destination and he let out a quick, "Okay." Honestly I think I may have spooked him a bit and I apologized for getting worked up but that it is a touchy subject with me. I gave him a tip but a low rating.
Why do these people insist on being so persistent? I often tell them that neither of us are going to change the others' mind so why bother. This is why I don't post political or atheistic stuff on FB.