ask vs wonder

No, they are not interchangeable.

First of all, the word "ask" is either transitive or intransitive. If I ask my daughter if we have any milk, my daughter is the direct object. "Wonder" is intransitive. It doesn't take an object.

Second, "ask" is something active. It is a form of communication, usually either by speech or by text. Wonder is generally an internal mental state (something that I am thinking passively).

If Adam wonders about milk, he may say nothing. You might not even know what he is wondering.

If Adam asks about milk, then he is speaking to you in the form of a question about milk.

The confusion is that sometimes we use the word "wonder" is part of a dialog. This isn't the normal use of the word.

Thanks a lot for your reply.
What confused me much is finding some examples using "wonder" in reported questions:
"Who are you waiting for?" __ Alex wondered who I was waiting for.
(Oxford Guide to English Grammar. Page 354)
"Did you study hard for the exam?" (She wondered .....)

I was wrong about the intransitive part. "Wonder" is transitive in sentences like "I wonder what I will have for dinner".