I often design UIs in Google Docs (b/c it's really easy to share and let people comment).
I can't find a list of image (.png, etc.) versions of Font Awesome icons.
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Sign up to join this communityI broke down and made Font Awesome vectors in Google Draw, here you go:
https://drive.google.com/previewtemplate?id=1S_wrhxJ-8f9nlz1EZx75Ldl_EM60RdGEVSpi5CFEqTE&mode=public
Here's a write up of how to use them in Google Slides, should apply to Google Docs as well. http://phdesign.com.au/general/adding-font-awesome-icons-to-google-slides/
Basically you need to find an SVG (or vector graphics version) of the icon, convert it to WMF (e.g. using CloudConvert) then you can open it in Google Drive and copy and paste it.
There is an updated add-on for Google Slides, it's called Insert icons for Slides by Romain Vialard
see https://gsuite.google.com/marketplace/app/insert_icons_for_slides/96201000411
For Google Sheets it is not possible to add icons directly into the text, but you can add HTML entities and symbols. Just copy and paste from here: https://dev.w3.org/html5/html-author/charref
there is a fantastic add-ons now, search for awesome, it is named "insert icons for doc". see https://chrome.google.com/webstore/detail/insert-icons-for-docs/ihdckhgffclpplbgdlhmbdhmjmlhpbnd?hl=en