Genesis 1:2 "and the earth being without form and empty, and darkness on the face of the deep, and the Spirit of God moving gently on the face of the waters," LITV
Genesis 1:2 "And the earth was without form, and void; and darkness was upon the face of the deep. And the Spirit of God moved upon the face of the waters." KJV
The word gently is an extra insertion into the text. H7363 is the Strong's number for "moved" in this verse. Perhaps one reason why J.P. Green chose to translate H7363 as "moving gently" is that the definition for H7363 is "A primitive root; to brood; by implication to be relaxed." So he may have seen that last part of that definition and wanted to reflect that idea in the translation. This idea is expressed more fully in the entry for Gesenius' Lexicon.
Also, our modern usage of brood usually thinks about the usage of brood in the sense of "To remain a long time in anxiety or solicitous thought; to have the mind uninterruptedly dwell a long time on a subject; as, the miser broods over his gold." The main meaning of brood is "To sit on and cover, as a fowl on her eggs for the purpose of warming them and hatching chickens, or as a hen over her chickens, to warm and protect them."