There were conflicts between the Catholics and the Protestants, especially when large numbers of Irish Catholics came across during the potato famine. These mostly would be things like discrimination against the Catholics and occasional riots. The Know-Nothing Party was an American anti-catholic and anti-immigrant nativist political party that started several anti-catholic riots across the USA in the mid-1800's.

But by and large these riots occurred back east, and you specifically asked about the Wild West (the Wild West, by the way, never really existed like it does in your imagination). For that we need to look at the Mormons.

The Mormons were (and are) a sect started by Joseph Smith in the 1820's. The began spreading westwards, to Ohio and Missouri. While attempting to organize they had several conflicts with the existing settlers, culminating in the Mormon War of 1836, when the governor of Missouri declared that the Mormons must be exterminated or driven from the state. They relocated to Illinois. In Illinois conflicts and tensions escalated until eventually in 1844 Joseph Smith and his brother were killed by an anti-Mormon mob. The Latter Day Saints over the next 2 years moved to Nebraska and then eventually outside of the United States altogether, in a place that would later become the Utah territory. But the USA expanded. Other non-Mormon settlers moved into Utah and conflicts escalated, especially when the Mormon practice of plural marriages became known. Eventually things escalated to the point where the president of the USA James Buchanan in 1857 sent the Army to Utah in what became known as the Utah War of 1857 and 1858. The highlight of this war was probably the Mountain Meadow Massacre, when Mormons slaughtered 120 members of a wagon train travelling through their territory over a 5 day period and tried to blame it on Indians. The war ended when the leader of the Mormon church, Brigham Young stepped down.

But relations continued to sour. in 1878 the US Supreme court ruled that plural marriages were illegal and the US government began seizing church assets. In 1890 the Church of Latter Day Saints officially disavowed plural marriages, and relations began to improve, to the point where in 1896 Utah was accepted as a US State.

So there's an example of the US government going to war against a religion in the Wild West. Also during this time anti-catholic sentiment was widespread, but fairly subdued in the territories out west, it was more of an Eastern thing.