G Reco takes place 1000+ years after Universal Century ends.
So it's the same universe, but everything and everyone from the original UC shows are long gone.
The story goes that at the end of the Universal Century, the powers that be went to live at the moon, and placed a ban on technology on Earth and took control of the management of resources. So that war and the atrocities committed in the UC would never happen again, and so that the Earth may heal and recover.
G Reco starts with factions among the moonfolk being tired of living "exiled" in space while the Earthlings get to live in what is now paradise. So they try to reconquer Earth (which is why it's called Reconguista, from the spannish word Reconquista meaning reconquer) by sowing seeds of discord and creating a civil war on Earth.
Meanwhile the main protag just travels around in space with his space pirate crew killing hundreds of people while shouting ME NO WANT ANY TRUBBURU.
(http://i.imgur.com/lPLYvLc.jpg)
She stops being retarded halfway through. Then she goes on to pilot this sexy thing:
(http://i.imgur.com/TAy6pdz.jpg)
I love you, G-Lucifer!
(http://i.imgur.com/xigiBja.jpg)