Galactic Federation Headquarters is the primary base of operations of the Galactic Federation. Not much else is known about the HQ except that, according to the NES Metroid manual, it is located somewhere on Earth and is also the headquarters for the Federation Police. Samus's X Parasite surgery occurred here, and it was referred to in the manuals of Metroid II: Return of Samus and Super Metroid as the destination that the ambushed research SR388 crew of the Space Research Vessel Marina were attempting to reach.
In Metroid: Other M and the Metroid: Other M Art Folio, it is revealed that Samus spent some of her time training with the Galactic Federation under Adam Malkovich here. It appears to be floating atop a man-made island, thought not far from land.