No, Herbert Hoover and J. Edgar Hoover were not related.
Herbert Hoover (1874–1964) was the 31st President of the United States, serving from 1929 to 1933. He was a mining engineer and businessman before entering politics, and his presidency is often associated with the onset of the Great Depression.
On the other hand, J. Edgar Hoover (1895–1972) was the first Director of the Federal Bureau of Investigation (FBI), serving from its founding in 1935 until his death in 1972. He is known for his controversial methods and for significantly expanding the FBI’s role in law enforcement.
While they share the same last name and lived during the same time period, there is no familial connection between them. Their careers and contributions to American society were distinct and separate.