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Hi @Gael
Thank you for posting your question in the Microsoft Q&A forum.
After reviewing your post, I test it on my side and there are a few things I noticed.
I'll get the globe icon when adding a page from another site. (a page in Site A when add to Site B will be displayed as a globe).
May I ask how you add a site in Quick Link? By using Sites or From a link?
Every link from a different site will show up as globe icon because Microsoft cannot detect even if it's in the same tenant.
However, I found a workaround for you. When you inspect an element, you can point to SharePoint logo, it will return a link like this.
https://res-1.cdn.office.net/files/fabric-cdn-prod_20240610.001/assets/item-types/64_1.5x/spo.png
Save the file and upload it to OneDrive. You can edit the file logo by Edit Quick Link.
I know this is not yet the best solution because you have to do extra work.
Thanks for your understanding.
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