"Did you have your breakfast" is correct. You can also say "have you had your breakfast?" and it means the same thing. In this context, "did" and "have" are helping verbs that make the sentence past-tense, and the verb that tells what is happening ("have") is in present tense.

The correct is "Did you have your breakfast?". The reason is that after "did" your verb goes in present: * Did you eat? * Did you do your homework?

did you have your breakfast