I dont think it is possible to do this by using the contact form 7 plugin. Because the plugin uses bind('click') on the submit button JS trigger, where it supposed to be live('click') to work on an AJAX loaded form.
As an alternative you can use a custom made contact form, or hide the contact form by default using JS and show it on a click of a button. The second solution is easier and still works without JS.
Hope this help