This question cuts two ways. Can chatbots be trained from Quora data? And, can chatbots be used to ask/answer questions on Quora? There has been a lot of discussion about both these issues on Quora already.
AFAIK, there is no public dataset of Quora questions and answers available per se. Also, there is no proper/official Quora API available in order to connect chatbots with Quora. There has been a lot of speculation about why the powers that be at Quora are lagging behind the parade in this respect.
See also my Quora answers to: