What coordinate system is at rest relative to the center of the galaxy? I am given equatorial coordinates and I need to transform them into a coordinate system that is at rest relative to the center of the galaxy. The closest thing I could find is galactic coordinates which seems like they are fixed with the sun at the center. Wouldn't this meant that they are moving with the same velocity as the sun and they are rotating?
Galactic coordinates are sun centred with a zero longitude chosen to align with the centre of the galaxy when the measurements were made in 1958. 

As defined, the galactic coordinates don't rotate, and so the position of Sag A* is very slowly drifting. Currently Sag A* is about 4 arc-minutes from the defined zero longitude, but decreasing by 5.7 milli-arcseconds per year (a tiny amount)

The task is ill defined, You can convert between different coordinate systems of the sky, but they all must be centred on the solar system.