It depends on how your "network path" actually works (though it sounds like Samba, or some other zeroconf protocol).
Git can only operate with HTTP(S) and SSH protocols as transports, unless you configure one of those properly, you can't use git on your router as a remote.
Read http://git-scm.com/book/en/Git-Internals-Transfer-Protocols for more info. (Also, this question has sort-of been answered before Using Git with a Samba shared folder).