We do store referrer, but not in a way that's easily or sanely accessible from the Q&A sites (it's just in the traffic logs) nor is it necessarily connected to the badge awards (the badges are per-IP, not per source, so outside of curiosity we don't really care where people came from so long as they did at all).
This would require building out something new to track and display this info when it's available at all, and given the current badge numbers on SO (vs the actual # of visits/page views we get per day), I'm not sure it's all that worth it in terms of either dev time or storage cost.
Having said that, in the case of your recent badge, I'm afraid there's nothing conclusive to say here. I figured I'd look up referrers on the day the badge was awarded... and they're all either NULL or Google search sites (and one that looks like a Google+ search? I'm not sure how that works; I don't use G+.). This is unfortunately gonna have to remain a mystery. ¯\_(ツ)_/¯