This is not a feature of the Contact Form Solution Pack or any of our other extensions. The captcha image is loaded when the page is loaded, in order to get it to update you can reload the page but this will clear out any of the already entered values. You could do some custom ajax coding to get the images to update also.
If you have cookies toolkit you could use a combination of cookies and js to do this. You could have a image or link to reload the captcha that would trigger a 'store form in cookie' behavior, then reloads the page. When the page loads you can populate the form from the cookie.
These techniques may be a little advanced depending on your experience and skills but if you would like to try this with cookies toolkit I can give you some pointers.