Hello @freedomdesign,
Thanks for updating the thread. 
I suggest you to take child theme route to implement Google Adwords conversion code. Please page the following code in child theme function.php file. Before that please refer following resources to download and install Child Theme.
https://theme.co/apex/child-themes
You can take a look at following post to locate Page ID. Though the article talks about post id but essentially logic remains the same.
function my_custom_tracking_code() {
if( is_page(185) ) {?>
<!--replace this line with your Google adwords conversion code -->
<?php
}
}
add_action('wp_head','my_custom_tracking_code');
You just need to replace 185
from above code with the page id of Contact us page. And <!--replace this line with your Google adwords conversion code -->
with conversion code.
Please let us know how it goes.
Thanks.