Does the galaxy move in a circular motion?

Do the quadrants of the galaxy move with the section of the galaxy they describe, or does the galaxy move through each quadrant?

Is there a such thing as a galaxy year?

"Does the galaxy move in a circular motion?"
Yes.

"Do the quadrants of the galaxy move with the section of the galaxy they describe, or does the galaxy move through each quadrant?"
The quadrants move with the section they describe, otherwise they would have no meaning.

Is there a such thing as a galaxy year?
Yes. It takes 226 million years for one complete rotation.