This is a great question! The answer is yes- let me explain why.
As you are sitting stationary in your chair, it feels like you are not moving but you are since the earth is rotating and orbiting our sun. If you could sit on the Sun, then you would feel stationary but you would actually be moving since our sun is moving within our galaxy. The Milky Way Galaxy is a spiral galaxy made up of many suns which each may have their own separate solar systems. Thus, with the spinning of our galaxy and the expansion of the universe, yes our Sun does move within its own plane of reference when compared to our galaxy.