Is the game done if your only movement is getting out of check? If I play as White and use my queen to check Black on every move, and Black can only move their king to get out of the check, can I win the game?  White could move something else, but they decide to check on every move.

This happened years ago when I was new to the game in a tournament, and the referee gave me the win as it was not checkmate, but my opponent only had a king and two pawns left. Was that my win?
No, this is not a win.

What you describe is called perpetual check and according to the rules of chess it results in a draw.