How to disable quickedit for specific block, but keep contextual links
It looks like you can use module https://www.drupal.org/project/block_content_permissions for that. To cite the project page:

Add the following permissions to allow access to edit blocks in-place (on-page):

Use contextual links (Contextual Links module)
Access in-place editing (optional, Quick Edit module)


However, it might not be feasible depending on the number of blocks on you site.