Let's make sure you are asking the right question. Game designers create the mechanics and systems of games. Maybe even the story, too. Game developers usually refers to the whole team or the programmers specifically.
So if you want to be a game designer, there are schools that offer degrees in that, but most designers majored in economics, history or other liberal arts. You need a broad education to draw upon for your designs. It is then on you to play as many games as you can (digital and tabletop) to learn about the mechanics and how they can be used.
If you actually meant game developer, then any school with a strong computer science degree will serve you well. You can learn the specifics of the game you are working on as you go.
Be careful. Many schools get this terminology wrong too. Many say they offer a game design degree when it is actually game art or game development.