Beta sites aren't part of Data.SE, however if this site wasn't a beta site you could track user participation in the site through there. I even have some queries saved in an answer of mine on Programmers that I often re-use to check out a site's user participation trends.
However since this is a beta site, the only stats we have access to are the ones on Area51. I know some users are actually tracking those stats on a regular basis and graphing the trends, so perhaps ask in chat if anyone has links to these trends. normalocitynormalocity is regularly tracking the site's stats, you can see the latest stats on his site.
There's also Quantcast data, but that only measures visitors and views, not necessarily site participation.
It should be noted that none of this is really tied to a specific question. You can make assumptions that the traffic changes were the result of a really popular question, however its not something that I think we can actually prove.
(As a side note, I think the question you linked is on-topic after the latest edits)