how can I use special Font awesome character in Twig link()?
The point of the FontAwesome selectors is that you do not need to print out the Unicode equivalent for the character. FontAwesome usually works by adding a pseudo element before or after the HTML element using the class.

In this case, use the fa fa-envelope classes and optionally add in a class to hide the text with a text-indent or something.

{{ link('Contact Us', 'base:contact', { 'class':['fa', 'fa-envelope', 'fa-lg', 'contact-us-link'] }) }}