IntelliJ Idea - search for a value in debugger

Sorry to disappoint you, I don't think that this option exists. The quick search will only look for values in expanded objects.

You can use some other tool to create a memory dump of this object, and look for your data there.

Expand this tree so that you see all those strings, and start typing what you search for. The speed search should do the rest. To get to the next/previous results press the down/up arrows respectively.