Is there an open source platform that allows a developer to quickly build a Q&A site? For example, was StackOverflow built from the ground up or did it leverage an existing platform?
As far as I know Stackoverflow was built from scratch using .net
Here is a presentation at Google that explains the ideas behind stack overflow
http://www.youtube.com/watch?v=NWHfY_lvKIQ
There was a person on SO some time ago who wrote a question, asking a similar question and answered it saying that he had built something like SO and made it open source - it was/is called Stacked.
http://ra-ajax.org/stacked-an-open-source-implementation-of-stackoverflow-com.blog
SO was built from scratch in .net. The guys have discussed releasing the SO code as open source, but have not come to a decision yet. (see here http://stackoverflow.com/questions/7335/the-stackoverflow-source-code and here https://stackoverflow.fogbugz.com/default.asp?W4331).
There are various open source copies around you could look at. Ankur mentioned one. There is also cnprog (here http://code.google.com/p/cnprog/)
DotNetNuke is great. It has a whole lot of modules that could be used or adapted to do a Q&A. And since it is ubiquitous, there are lots of resources (including SO) where you can get help.