"Can I have a question?"

Is this phrase used at all by native speakers? After almost 20 years of studying and speaking English, I've been told this is a phrase that almost every Czech speaker gets wrong - it seems that even my university english teachers got it wrong, which is mindblowing to me. The fact that I did not get this corrected by anyone during my whole life makes it very surreal - "Can I ask a question?" sounds correct to me, however "Can I have a question?" feels better.

Is this phrase really incorrect? How foreign does it sound to a native speaker?

No, this is not a phrase that any native speaker would use, and yes it is incorrect. One would either say "I have a question" or "May / Can I ask a question" instead.

"Can I ask a question?" is commonly said, but better still is "May I ask a question?" "Can" generally refers to having the ability to do something, whereas "may" is a way of asking permission to do something. This is a fine point that is becoming less of an issue in English speaking (many avoid "may" because it is also used to mean "perhaps," and in certain contexts could be confusing), but I've never heard any native English speaker say, "Can I have a question?"