Github is the most popular online service for hosting and collaborative development of open source projects. And Redmine is the most popular personal and enterprise open source project hosting and management solution. So, no wonder, that these two systems are used together quite often.
Thus, Github’s Redmine service hook can be used to trigger fetching of repository updates by Redmine, when new commits are made to the Github repository. Unfortunately, it’s useless for the pure Redmine, as Redmine is not able to work with Github repositories directly (only through the local Git mirror, which needs to be created and updated manually)… Moreover, I am not aware of any solution, which could be used for this (even through the mirror).
Certainly, Redmine (especially its community) does much more to support Github. Thus, there is the plenty of tutorials describing how to mirror and keep Github repositories in sync for Redmine (e.g., this or this). Also there exist special plugins for making Github repositories easier usable under Redmine. The most noticeable such plugin is Github hook authored by Jakob Skjerning.