He was the president of the Utah State Senate from 1965 to 1966, subsequently becoming the chairman of the law firm of Kirton McConkie.

He grew up primarily in Salt Lake City, but, as a toddler he lived in Ann Arbor, where his father was pursuing a law degree at the University of Michigan.

Oscar, Sr., was a State District Court judge for many years and was the Democratic candidate for governor of Utah in 1960.

He then went to the University of Utah, where he received a bachelor's degree in 1949 in political science and a Juris Doctor in 1952.

[11][12] McConkie argued the case that led to a broadening of the clergy-penitent privilege in Utah to include any information that was given by a parishioner to a cleric with the intent of getting spiritual advice.

McConkie was a key figure in the process of getting official recognition of the LDS Church in Zaire in 1986.