A couple of comments...
1. You correctly pass in season and episode as parameters, but then you aren't using them.  The data initialization should contain "season: season, episode: episode"
2. Progress is not an attribute of show.  It should be another parameter.
3. plugin_version is not an attribute of the Client class, but it looks like (based on my reading of the API), plugin_version should be set to the version of trakt.  That should be defined as Traktr::VERSION (you may have to require 'traktr/version')
4. media_center_version and media_center_date are not attributes of the Client class, and they don't really belong there either because the Client class is used to connect and authenticate to Trakt, not to describe the media center using this library.  They should also just be passed in as parameters.