Is there a way to integrate Geany to Github? How it's done?

Their documentation seems to be lacking information on this. At geany-plugins I see GitChangebar, but I failed to find instructions.

Are there any ways of managing commits from geany?

You can use the plugin GeanyVC to commit and do some more basic tasks on your code. GitChangebar is only able to show you changes done compared with current HEAD of your working copy. Currently -- Feb 2016 -- there is no official plugin available to push/pull or do branch management.

GeanyVC and GitChangebar is a good addition for command line you could use via VTE at bottom of Geany's window.