When you say you want to use Google Adwords tracking code, I'll assume you mean Google Analytics generally.
You can track page loads to the contact page with a specific goal set in Analytics specifying the page URL. In order to capture successful form submissions you will either need to redirect users to a success / thank you page and track that page loading as a goal.
Better yet, you can set up Google Tag Manager with an event for form submission success and you don't need a thank you page. Neither of these solutions is internal to Magento 2 other than the basic tracking setup you probably want in place regardless of the specific tracking request.