There is no automated way to do this, but if you want to do this with git, I would recommend exporting your database in settings and re-importing it when the data is updated. It has a replication model so you won't lose your data added after your last commit (new cards, jumbles), but any conflicts like edited cards on both repos will just resolve themselves to the newest version. It should also work with Dropbox, Google Drive and other fanciness, but git is probably better as it has a more transparent file history.
Not all data is replicated; sketcher images, docker and todos on the homepage are stored outside the main db. Also bear in mind that all the kanbans and jumbles are exported/replicated, potentially including your passwords or some sensitive information.