Yes. JOHN SEVIER, a Representative from North Carolina and from Tennessee; born near Harrisonburg, Rockingham County, Virginia, September 23, 1745; was elected Governor of "the proclaimed" State of Franklin in March 1785 and served for three years. He was in fact the first governor of Tennessee.
William Blout was the governor of the Southwest Territory from 1790 - March 30, 1796. On June 1, 1796 the Southwest Territory became the State of Tennessee and was admitted as a state in the United States. John Sevier served as governor of Tennessee March 30, 1796 - September 23, 1801.
Between the time that North Carolina surrendered its western part to the U.S. government and the time that the land was admitted as the State of Tennessee, William Blount was the Governor. The first Governor of the State of Tennessee was John Sevier.