I'm not good with all English dialects. I see that Merriam-Webster lists the meaning of "to ask a question of or about", for which "a întreba" would be appropriate.
However, it wouldn't work for the meanings "to doubt" and "to interrogate", which should be translated as "a se îndoi" and "a interoga" respectively.
"Ask" is the most common term in English for simply posing a question. As a verb "to question" can have some negative connotations. "to question someone's sincerity" is to doubt them. When the police question someone it means "to interrogate". And "to put someone to the question" is a euphemism for "to torture".
So the most neutral expression would be "to ask a question" or "to pose a question". where "question" becomes a noun object of the verb. So, in a classroom. both teachers and students will "ask questions" not "question".