You could ask 'what order', but I would probably just guess any position and say 'were you first?' or 'are you eldest?' and usually a native speaker would fill in the details, of 'oh no, I'm third, my sister...' It depends slightly on the situation and if it is a casual situation or more formal.

"What is your birth order" is a standard question and all that is required. By the way "queue" does not fit your question. "position in birth order" perhaps

I guess you could say this: Where are you in the birth order of your siblings?

What do you mean by 'what's your queue among siblings?' Do you mean you want to find out how old they are compared to their siblings? Then you can say, "Are you the third child?"