When I search for my clients church website, google shows the Rock favicon instead of their favicon (Site icon).
First check if the Site icon has been set. In Rock go to CMS Configuration > Sites. Choose the site and see if there is a Site icon set. This is where the favicon is pulled from.
If this has been set you may still see the Rock favicon in a search engine. Google will eventually reindex the site and this icon will be refreshed.
The site owner can force an update in Google Search Console by requesting a reindex. https://support.google.com/webmasters/answer/9012289#request_indexing