In the bottom right of the SQ project overview page is button labeled "Get Project Badges".

From there you can select a number of different badge metric options, including "Quality Gate status". Copy the url and paste to your dashboard.

There is just one reference in the documentation to this ...
"How can I promote the health of my project to peers ?"
The caveat is it seems to require the project be "publicly visible", that is you do not need to explicitly log in to SQ to see the projwct overview page. No word if it works w/tokens, etc. SQ security model is rather "chunky".
Note there may be an issue with the widget