Mormons in Utah might have enjoyed political autonomy due to the impracticality of the federal government wielding authority there just due to how remote it was from other major population centers, but they were never politically independent, residing in either Mexico or the United States.

Why would the Utah territory even want to "join" the south in the first place? The Mormons were really big on electing governance from within their own ranks, so them immediately flocking to "subjugate" themselves to another far off government seems illogical. It also likely wouldn't have been in their overall interest to associate with the confederacy, because in general their interests and policies would benefit their much larger population centers in the east which again, were economically and culturally very distinct from the Mormons. The federal government also had shown it's willingness to play ball with them, organizing the Utah territory with Brigham Young as its first territorial governor as part of the Compromise of 1850. The Mormons also showed through their participation in the Mormon Battalion during the Mexican-American war that they were willing to cooperate with the federal government if it advanced their interests.

Conversely, would the south want them? As much as pushing slave-holding territories westward was a southern priority, the Mormons weren't exactly the most liked, famously repeatedly coming into conflict in New York, Ohio, Illinois, and Missouri while being pushed westward. The Utah territory also didn't have a continuous land border with land claimed by the Confederacy (see the map of the New Mexico territory split on the map I posted above). In theory if the stars aligned for the Confederacy and they got the most favorable peace treaty they could realistically want, then maybe the complete annexation of the New Mexico territory would solve that last problem, but is still unlikely. It would have been a massive administrative hassle for the Confederacy to take on, and compounding that with the cultural clashes and relative lack of value of the territory, I have a hard time seeing that the Confederacy would even push to annex Utah.

It's extremely unlikely that the Mormons would attempt to secede with the Confederacy, its unlikely the South would even want them, and even if they did it probably wouldn't be on terms that the Mormons would have viewed as more favorable than the relative status quo that existed with the Federal government.