If its just "XYZ just logged into ABC Q&A site" then all you need to do is take the stream_publish permission and post an update using the Graph Api.
However if you want to Publish all their activity like "Added a new question", "Answered a new Question", "Commented, Upvoted etc." - this can be done using the Facebook Open Graph plugin which involves some effort to get integrated around the Q2A site.
Steps I would take to do so:
1. Facebook Connect / Facebook
2. Take the Publish_Actions permission from user
3. Go through the simple Graph API Post method (Details here - https://developers.facebook.com/docs/opengraph/getting-started
Do let me know if you or anyone is looking for the code to do this. I have implemnted it in one Q&A and wordpress site, so I should be able to help you out with a detailed tutorial or something.