Database projects in Visual Studio have the features you need. You can compare schemas to track differences between schema versions. They can also handle deployment, although your deployment needs may vary.
@MSDN Database Projects
There are other third-party tools that can be used, such as SQL Compare by Redgate.
SQL Compare by Redgate
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Unfortunately, none of them have built-in support for git. But that would not stop you from overcoming this gap if you don't mind doing most of the work yourself.
With Visual Studio, you can use MSBuild to automatically run scripts / executables related to compilation actions. From there, you can attach your own interactions using git.
MSBuild Reference @MSDN
If you are looking for a tool that can automatically deploy a schema change from git, I do not know any existing at this time; but you can roll on your own using SMO. It has all the functions necessary for scripting SQL Server objects and scripting for SQL Server. You may have to run your own incremental scripts to modify tables (as a security issue), but you can do drop / replace for many other objects that are not data (stored procedures, user-defined functions, etc.).
SQL Server @MSDN Management Objects
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