It seems like the plugin is not working properly.
Please follow the below steps.
> Launch Lightroom
> Click on Go to Publishing manager.
> Click on Plug-in Manager button
> Choose Flickr from the List
> In the right hand side panel, expand Plug-in Author Tools
> Click on Reload Plug-in button
> Click on Done
> Try to Authorize to Flickr now
Let me know if this helps.
If it is still not working, please post a screenshot of the error message.