Here you can find extensive question lists for different purposes (Getting to know someone, Would you rather, Truth or Dare, etc.) It's great for when the creative juices aren't flowing. My favorite list is the Deep Conversation Topics.
This is an impressive site from the Internet TESL Journal. Here you can find question lists for an incredible amount of topics. Also, check out the other sections in the menu bar. There's a lot of great ideas and resources for teachers.
I looked for a while and this was the only website I could find like this. It has a random question generator that is good for a conversation class. I find that it works best with intermediate students (A2-B2). It seems that the most difficult questions tend to be second conditional or present perfect.