There was never any question that the Utah Territory would remain on the Union side. In 1857-1858 (a few years before the American Civil War started), President James Buchanan dispatched US federal troops to Utah to put down and destroy an alleged religious dictatorship. He installed as Utah Territory Governor Alfred Cumming to replace the Mormon leader Brigham Young.

However, Cumming left Utah quietly on 17 May 1861, following the Confederate Army attack of Fort Sumter, South Carolina the month before. General Albert Sidney Johnston, another leading figure in the territory, left the area during the same period. Both were Southerners and chose to return to the South rather than remain in a territory that would remain under Union control.

The first message from Salt Lake City on the newly completed transcontinental telegraph in October 1861 came from Brigham Young. His message was to President Lincoln: "Utah has not seceded, but is firm for the Constitution and laws of our once happy country."

During the war, the Utah Territory provided a cavalry company to the Union forces, led by Captain Lot Smith, who had his commission from the United States Army. Smith’s company was composed of approximately one hundred six soldiers that included officers, privates, and teamsters. Their service was for ninety days. The soldiers commanded by Lot Smith qualified for federal veterans’ pensions after the war. The company saw no battle and guarded a portion of the Overland Trail.

To the Latter-day Saints ("Mormons"), the Civil War was a fulfillment of a series of prophecies from their founder Joseph Smith Jr. During the war, Brigham Young and other church leaders at the time proclaimed that it was retributive justice against the US government and its people who had sanctioned the expulsion of Mormons from their homes in Missouri in the 1830s, their persecutions in Nauvoo, Illinois, and the martyrdom of their prophet-leader.

References:

(1) Civil War Saints, ed. Kenneth L. Alford (Provo, UT: Religious Studies Center; Salt Lake City: Deseret Book, 2012), especially the chapter by Richard E. Bennett, "'We Know No North, No South, No East, No West': Mormon Interpretations of the Civil War, 1861–65," pages 93–105. Available online: https://rsc.byu.edu/pt-pt/award-winning-recent-books/civil-war-saints

(2) Brigham Young, in Journal of Discourses 26 vols. (London: Latter-day Saints’ Book Depot, 1854–86), 8:324-325, February 10, 1861.

See also:

(1) E. B. Long, The Saints and the Union: Utah Territory during the Civil War (Champaign: University of Illinois Press, 1981)

(2) Gustive O. Larson, “Utah and the Civil War,” Utah Historical Quarterly 33, no. 1 (Winter 1965): 55–67

(3) Ray C. Colton, The Civil War in the Western Territories: Arizona, Colorado, New Mexico, and Utah (Norman: University of Oklahoma Press, 1959).

The Latter-Day Saint denominations and their relations with black people have always been...complicated, to put it best. When the Civil War broke out, slavery was perfectly legal (if not very common) in the Utah Territory, and Brigham Young was on the record with many statements regarding the natural supremacy of the white population and the exclusion of black men from the priesthood, on account of how black people were allegedly descended from Ham (who was cursed) and his wife, a descendant of Cain (who was cursed). So Utah was not exactly a hotbed of abolitionism, although its economy was not heavily reliant on slavery and when the Confederacy was formed, Utah, like most of the other territories, remained under Union control more or less by default.

At this point in time, there was very little love between the Mormon community and the federal government of the United States -- apart from the long-standing concerns over polygamy and the division of western land among different settler groups, the LDS Church had gotten into an actual shooting war with the army, a brief conflict in the late 1850s that basically ended when James Buchanan said "everybody is forgiven because holy hell we have bigger problems to deal with right now." There were also federal soldiers posted in Utah at around this time, just to make sure that any further skirmishes with political dissidents (who were mostly still alive, on account of President Buchanan's amnesty) didn't get out of hand. The outbreak of the Civil War made the United States government realize that it had an acute shortage of trained soldiers, so the vast majority of its forces in the territories, including Utah, were withdrawn in order to face the Confederacy and reinforce the many volunteer units that were now being raised to make up the manpower shortfall. At least some of the men posted to Utah deserted in order to join the Confederacy, but these were southerners serving in the army and not Mormons from Utah, at least in most accounts I've read.

The Civil War, as we think of it as a conflict between the north and the south, did not extend to Utah. Rather, there was a continuation of the intermittent raids and battles between American settlers and the indigenous nations (particularly the Shoshone) which intensified as the U.S. Army was suddenly distracted in 1861. Further information on the composition of American military forces on the western frontier may be found in other sources, but an interesting wrinkle emerged during the Civil War with the advent of the so-called "Galvanized Yankees." A certain number of Confederate soldiers, having fought and been captured by the Union, came to realize that they didn't like the Confederacy all that much and subsequently volunteered to fight for the Union. Because their reliability was considered suspect when used against their former comrades in the main theatres of the Civil War, their primary use came to be in the western territories (including Utah), where their presence helped alleviate the need for regular soldiers in the usual garrisons, and freed up units that could be brought to bear against the rebellion. This is touched upon in Bell Irvin Wiley's seminal text The Life of Johnny Reb, published 1943, which was the first really comprehensive text on common Confederate soldiers and which employs considerable primary source material. I highly recommend it for questions on the life of ordinary people, particularly soldiers, in the Confederacy.