MONTICELLO, Utah—Navajo people managed their affairs for centuries in the spectacular buttes and mesas of what is now San Juan County, Utah, until white settlers moved in and established political and economic control here nearly 150 years ago.

Now that the Navajo have taken over the county commission for the first time, a political battle is breaking out as they push for major changes, including opposition to the Trump administration’s policy to allow more development on federal land. In response, a politically conservative...