I gather that the company that runs the Stack Exchange network of sites has a role with this title.
What is this role about? Is the job description published anywhere?
It seems that "Public Q&A" are the question-and-answer sites accessible and usable by the public, as opposed to (edit:) Stack Overflow for Teams where only a limited set of people can become members. This includes several tiers - Basic, Business, Enterprise, where the latter is not hosted together with the public sites. Teams would be "Non-public Q&A" I guess.
So, it seems the Director of "Public Q&A" directs the development, design and engineering efforts of the entire public Stack Exchange network, and answers to Mary Ferguson, VP of Engineering; who in turn answers to David Fullerton, the company CTO.
In the context of the recent, uh, events, that means, in particular: "Public Q&A" does not include public relations, engagement with the public, being the public face of anything, or making SE-Inc-held information public which wasn't that way before.
See also the blog post Meet the Public Q&A team linked to by @PeterMortensen.