When choosing CAPTCHA Q&A (not sure if the other CAPTCHA options are the same, not used for the contact form) please make it possible to use CAPTCHA Q&A with the contact form as well.
If Q&A CAPTCHA is enabled, and a question set, guests will see and have to do the Q&A before submitting. Registered users, etc., will not be challenged with the Q&A.
Are you logged in? If so, log out and try the form again. If it's not coming up, ensure Q&A is enabled and a question set. Retest it.