Like in the image, the publish & other commands are not enabled.
To sum up comments, answer to your question:
To connect to GeoServer from QGIS installed with OpenGeoSuite you have to install OpenGeo Explorer plugin first.
Before you can publish layers or styles, you have to connect to GeoServer instance by option: 'GeoServer catalogs'. When doing OpenGeo Suite default install, you should put in URL field:
Default login for GeoServer is
admin and password is