How can I search internal blocks in the admin section?
If the block you're looking for is a "custom block", i.e. one that you create via structure->blocks->Add block rather than one defined in a module then you should be able to query the database for it like this:

SELECT * FROM block_custom WHERE body like '%my-class%'


...changing "my-class" as appropriate.

Alternatively, you can inspect the markup of the block on the front end and look for a class such as block-123, then navigate to /admin/structure/block/manage/block/123/configure, replacing 123 with whichever block number you found.

If there's no body field on that page then it's probably a block defined by a module.  Then you can try to find a clue as to where it's defined in the classes on the contents of the block, for example if you see lots of classes with the word views in it then it will be a block defined in views and you should look in the views UI to find where it's defined.

Also, turning on the core contextual links module may help as well.

n.b. All this is based on the assumption that we're talking about Drupal 7.  Much of it applies for other versions with some tweaking.