The WordPress Google Forms settings page makes use of tabs to show information about the plugin and set its options. It appears that the theme I was using, which I am in the process developing, is the culprit. It loads some new jQuery functionality from Google’s API that apparently allowed the tabs to work as I expected. I am not sure yet why the builtin WordPress jQuery functionality isn’t working correctly as I expect it should.
Now that I know where the problem is, I think it should be pretty straight forward to fix it.