The answer is: no. Although he shares the same last name as President Herbert Hoover, who was in office from 1929-33, they are not the same person, and they are unrelated.
The link between President Hoover and J. Edgar Hoover began with President Hoover's insistence that both the Justice Department and the Internal Revenue Service (IRS) should go after 'gangsters' for tax evasion. This plan, along with the authority granted by President Hoover, led to the successful prosecution of the infamous mobster, Al Capone.
In the mid-1920s there were already an estimated 1,300 gangs in Chicago, alone, operating in full view of the American criminal justice system. Buoyed by a great deal of profit from bootlegging, as well as nearly-unchecked power in the cities, American gangsters were rampant. Rival gangs were infamous for violent, blood-soaked gun battles in the streets of their adopted cities, with more than 12,000 murders every year (by 1926). Law enforcement officials were literally 'outgunned' and outnumbered, having been overwhelmed by a surge of new crime including bootlegging, speakeasies, kidnapping, auto theft, gambling, bank robbery, and drug trafficking. Since 1929, the Bureau of Prohibition – headed by agent Eliot Ness – began an investigation into the business and personal life of Capone, in an attempt to convict him for violating Prohibition (the Volstead Act).
Following President Hoover's instructions, Frank J. Wilson went after Capone's violations of income tax law, leading to a 1931 indictment for tax evasion and a variety of Volstead Act violations. After a series of successful jury tampering incidents, the jury pool was unexpectedly switched with that of another case, preventing Capone from repeating the behavior. Finally, he was convicted and sentenced to 11 years in prison, along with hefty fines and liens on several properties. He was finally paroled November 16, 1939 and died after years of deteriorating health due to neurosyphillis, in 1947. As a direct result of President Hoover's instructions to J. Edgar Hoover, the method for investigating, charging, and convicting 'American gangsters' was finally perfected, leading to the downfall of a once-formidable empire, in 1938.
Besides Herbert Hoover, there were many presidents who came into close contact with J. Edgar Hoover. Because of the length of his service as director of the FBI – and as a direct result of his controversial career – the tenure of a director is now limited to one 10-year term, without congressional approval for extension. During his tenure as director, J. Edgar Hoover had a direct line of influence to eight U.S. presidents. Beginning with Calvin Coolidge, J. Edgar Hoover's presidential influence (due to his position in the bureau) lasted more than 40 years, through the presidencies of Herbert Hoover, Franklin D. Roosevelt, Harry Truman, Dwight D. Eisenhower, John F. Kennedy, Lyndon B. Johnson, and Richard M. Nixon. In fact, J. Edgar Hoover's controversial activities lasted the effects of several wars (World War II, the Korean War, and the Vietnam War), the Great Depression, and countless social and cultural revolutions in American society.