You cannot import the "relationship classes" because the ESRI FileGDB API can't read those unfortunately.
And the GDAL driver relies on the ESRI FileGDB API to get read/write functionality for a FileGDB.
But you can convert the FileGDB you have, only the feature classes and tables, to a PostgreSQL database with PostGIS.
The PostgreSQL database with the PostGIS extension allows you to store geometries with multi-user editing and access.
First, make sure your FileGDB is >= 10.0 because GDAL's ogr2ogr can't read from a FileGDB < 10.0. And you can't use any tools inside QGIS without GDAL having support for FileGDB formats. You can test this quickly by trying to load a FileGDB into QGIS - if it loads layers, it works, if not, then the version of GDAL that QGIS is running against doesn't support FileGDB - yet...
You should see it under the "Directory" type when adding a vector layer:
Second, configure and deploy a PostgreSQL/PostGIS database cluster/server to store all of this data.
Third, use ogr2ogr to translate the data from FileGDB to PostgreSQL. Inside QGIS there is also a number of options like right-clicking each layer from the FileGDB and doing a "Save as...", using the DBManager, SPIT Plugin (uses Shapefiles though).
Fourth, re-create your old relationship classes in the PostgreSQL database using the various SQL tools at your disposable - notably creating VIEWS from various types of JOINs. This is the part that can take some time depending on the type and amount of relationship classes your old FileGDB had.
You will definitely enjoy the fruits of your labour though and appreciate the multitude of skills you will acquire along the way (or just outsource). I didn't provide details of each step because they can change depending on the details of your source data. But this site as a whole should contain enough insight into each issue to get you started!
It will also be good to know that your data is now stored in a well documented Open SQL format that could set you up later on down the road to easily transfer that data to a different storage format.