The Federal Bureau of Investigation, also known as the FBI, was the primary law enforcement agency of the Federal government of the United States of America on Earth. The FBI was established in the first part of the 20th century, and remained active well into the 21st century.
Gary Seven presented himself as an FBI Special Agent in 1971, while investigating the deaths three years earlier of Agents 201 and 347. (TOS - Star Trek: Assignment: Earth comic: "We Have Met the Enemy...")
After a time traveling Pavel Chekov was captured on board the USS Enterprise (CVN-65), he was interrogated by an FBI "special field agent". After hearing Chekov's responses to his questions, the agent believed that while Chekov was a Russian he was also mentally impaired. (TOS novelization: Star Trek IV: The Voyage Home)
The FBI existed in some form in the 23rd century of the Kelvin timeline. In the year 2255 in that reality, the FBI unsuccessfully attempted to analyze two recordings of the serial killer known as the Doctor speaking in an unknown language. (TOS - Starfleet Academy novel: The Delta Anomaly)