It sounds like you want to change the git repository remote connections. When cloning on github to your local machine the remote connection in your local git-repo is the "origin" remote. You need to point that to Stash instead of GitHub. Other than that you should be fine.
git remote set-url origin "git://your.stash.repo.domain/yourproject.git"
Help on git remote: http://gitref.org/remotes/#remote