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.
I broke down and made Font Awesome vector graphics in Google Draw:
Google Drive Template Link: "Font Awesome Vector Icons/Glyphs"
Use Template
button to open the template in Google Draw.File
> Make a copy
or ALT+F CEdit
> Copy
or CTRL+CInsert
> Drawing
> New
or ALT+I D NSave and Close
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
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.
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