Is there a way to insert a CQWP to the page in SharePoint online public facing website?
According to Website feature comparison table Content Query Web Part is not supported in SharePoint Online Web Site.
But, you could consider the following approach:
- import CQWP web part into SharePoint Online Web Site (export it first, for example, from private site collection)
- Add exported web part on the page (see the figure below)
The following figure demonstrates CQWP configured to display Blog posts in SharePoint Online Web Site