There are two ways you can do this.
1) Quick, easy, free.
You can add a contact form to the quiz pass / fail results text. Most contact form plugins use shortcodes, and HD Quiz is fully compatible with shortcodes for the results text. If you are not using a contact form plugin, I recommend using my own HDForms, which is another of my free WordPress plugins.
+ free
+ quick and easy
– form only shows once quiz has been completed
– user is not forced to fill out the form
2) Save Results Pro Addon
You can also use the paid addon Save Results Pro to do this (can be found under HD Quiz -> addons). The addon has two options to show a form: before quiz or before results. Before quiz forces the user to fill out the form before being able to take the quiz, and before results allows the user to take the quiz, but then needs to fill out the form before being able to see their results.
– paid
– takes a bit more setup
+ can place the form before the quiz, or before user is shown results
+ forces all users to fill out the form
+ has a lot more features other than just the intake form