I'm a Git/Github newbie, forgive me if this is an elementary question, I might have not got the priciples right just yet (primary source: ProGit book). But I keep stumbling throughout the book, can't figure this out. Anyway...
You have your global .gitconfig where you already configured your SSH Keys/User Information. The global .gitconfig is overridden by a local gitconfig - the file "config" in your .git folder (if it does not exist you might have to create it). This website describes the process for Username and Emailadress but you can use every option avaliable.
For example you can copy the .gitconfig file into the .git folder (and rename it to "config") and just change the lines you want to change (probably github.user and github.token) or you create a new file with just the two lines in it.
If you prefer the command line "git config" you can avoid all the file moving stuff by omitting the "--global" option.