It means asking questions without prompting, simply for the sake of wondering and learning.

I disagree. What you are describing is one of the possible goals of intelligence, it doesn't define intelligence itself. Many humans are not really interested in wondering and learning, but we call them intelligent.

You totally can tune LLM that it will ask tons of questions to someone who created chat: how are you today? What are you doing right now? What are your hobbies? Etc.