Ms. Gaylor disputed suggestions that the foundation had unwittingly fostered the spread of the stickers. “I don’t think it has a thing to do with us,” she said.

Protests and warnings from critics like Ms. Gaylor also seem to be of little concern in places like Polk County, a few minutes from the Alabama border, where about 41,000 people live in a rural area dotted with churches, Confederate battle flags and fried chicken restaurants. The small atrium of Sheriff Moats’s building features a pair of murals painted by inmates, including one of the Ten Commandments on tablets that are more than six cinder blocks tall. A painted golden banner reading “In God We Trust” hangs above them.

The idea to add the national motto to patrol cars here, Sheriff Moats said, came after he saw on Facebook that Missouri sheriffs had begun doing so. Sheriff Moats spent $5 of his own money to buy a sticker for his department vehicle, which he said spurred deputies to ask how they might do the same. Within a week, most of the department’s cars had the stickers.

“I don’t know why an atheist is so upset about us putting up ‘In God We Trust,’ ” Sheriff Moats said. “I’m not saying that they trust God. I’m saying that we, as the guys in this department who put this on our cars, we trust in God. And why is that a bad thing? Even if you don’t believe, you know God’s all about good.”

He maintained that the motto’s presence did not signal that his department would discriminate. “You could be a satanic devil worshiper, and as long as you’re a law-abiding citizen and you need help, we’re going to help you,” he said.

There is nothing new about government display of the motto. The United States began stamping “In God We Trust” on some coins during the Civil War, and it has been on all coins since 1938. The words began to appear on paper currency in 1957.