Why does my G string sound flat?
The actual reason the G string falls out of tune most often and faster is because it's the string with the least amount of tension on it. However, if you tune it and it still sounds out of tune immediately, then that's a problem with your intonation.
Why does G string go out of tune on Floyd Rose?
It could just be the string stretching. Try stretching your strings by pulling up on the trem with the nut locked. However, in my experience most Floyd problems are from not using enough torque on the locks.