Quick links web part does not have settings for permission because all the permission settings in other locations still work for the links. If the users doesn't have the access to the page. They will be redirected to an empty page. They have no access to that page, so they can see nothing inside it. The permission settings always work.
Quick Link web part is simply a container holding links for users to get them easily. It does not and has no need to handle permissions. Permission settings are always needed to be done on the list/library or site itself.
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Hi @Jolyn Johnson
A url in list is a an item. If you want to use the url, you will need the permission to the library. Please check the other users have the permission to the library.
Thanks for the info, but why would they be able to see the URL and click on it, and then get the "You Need Permission to View the Library" if the URL is already part of the library?