We are being asked to vote on questions more often, however from a user’s profile page you can only see how many up and down votes they have done, you can’t see how the voting splits between questions and answers
Currently, You can't.
Maybe this is something the Stack Overflow API should address.
If you're polling your own vote information for the API, you should be able to get this information.
George is right, there's no way.
Well, there is a way, but it is more theoretical in nature. You could call
for every question, while providing your correction authentication information. That call returns a JSON object like this:
This is what my call for this very meta question returned. It means that I upvoted the question (38941) and George's answer (38942). VoteTypeId is 2 for upvotes and 3 for downvotes.
If you call it with a question ID, it returns both votes on the question and votes on answers to this question (which you can distinguish because it has a different number than the question). If you call it with an answer ID, it only returns the vote for this answer.
You should of cause be doing this slowly, so you don't get banned for excessive calls. Restricting this to questions in your active tags could at least give you an estimate on your question vote / answer vote ratio.
But then again, as I said, this is more theoretical.
Disclaimer: This answer is not endorsed (and is probably being frowned upon) by stackoverflow.com llc or U.S. Air.