Flickr integration was introduced in GNOME 3.10 when we added support for browsing Flickr content in Photos. In GNOME 3.12 we have taken this one step ahead. The Background Settings panel will now let you choose one of your Flickr pictures as your desktop’s background.
The panel only shows the 50 most recent pictures from each account to avoid overloading it with hundreds of items. If you are looking for that gorgeous photograph that you took a few summers ago, then you might want to use Photos instead. It has better search capability and is meant to handle larger data sets.