Since TeamCity 10.0 TeamCity integrates with GitHub, and the integration with GitHub issue tracker can be set up separately, or as a part of TeamCity integration with GitHub source code hosting service.
When setting up integration with GitHub, in addition to the repository URL and other general settings, you need to configure authentication and specify the issue ID pattern.
TeamCity allows you to select whether you want to connect to GitHub anonymously or to be authenticated via username/password or an Access token.
Converting Strings into Links to Issues
You also need to specify which strings should be recognized as references to issues in your tracker. For GitHub, you need to use the regex syntax, e.g. #(\d+).