The FBI uses an adapted form of Judo that was brought over from Japan after WWII. It’s evolved considerably since then, but some elements are still recognizable, particularly the throws.
It’s probably worth pointing out that this is what we’re referring to when talking about Police hand to hand. FBI instructors adapted Judo to fit the Bureau’s needs, and then shared that with the American law enforcement community.
I’m less certain what the CIA trains in. I suspect they receive training in one of the military forms (like MCMAP), but I don’t know specifically what The Farm trains CIA officers in.
For a spy, ideally, you’d learn local martial arts to assist in blending and protecting your cover identity. But, the CIA doesn’t really do that as much these days. They still maintain the concept of the Non-Official Cover (NOC) (a cover without diplomatic immunity), but The Agency as a whole is far more interested in signals intelligence than traditional tradecraft. Your more likely to see a CIA officer who is formally attached to the State Department, rather than an expat working as a tailor or running a restaurant. (Joking aside, service industry positions like those are very good places for spies to work undetected.)
The other thing worth remembering about the CIA is, when it’s planning operations that would be better suited to special forces units, they will actually use the appropriate units (like Delta or Navy SEALS), depending on what they need for the task at hand.
I’m sorry I can’t be more precise with the CIA, but they are a little evasive about exactly what they train their personnel in.