Why there is no article before words like queen and king?

I think this is to do with there only being one king or one queen.

'You will be prime minister' is another example where there is one prime minister - but 'you will be minister' where there are many ministers is not correct.

'you will be director general' - works too for an organization where there is one director general.

Now you can also say 'you will be the king', but 'you will be king' sounds better.

I believe this might be clarified with capitalization (though such capitalization is often optional). As tom noted in his answer, this tends to apply when there is only one such position in context.

You will be King one day.

With the capitalization, you can then imply the following.

You will be King Robert one day.

It's not so much that "king" is the noun, but the title. We can apply this further.

I want to be President when I grow up.

One of the examples you provided in response to tom's answer makes sense, too, and you might see it in Dan Brown's book, "Angels and Demons". (I'm constructing a sample sentence with that context.)

He aspired to be Pope, though he was ineligible as a candidate.

Where the noun can serve as a title, you may find it used without the article.

Regardless of what atrocities he committed on the way, he is Lord High Emperor; we have no recourse.

In the first two examples, both king and queen are used as quasi-proper nouns, meaning that they are similar to, but not exactly, proper nouns. As shown in this ngram, you certainly can capitalize the first letter to indicate that they are proper nouns. But that's not obligatory, because they're quasi-proper nouns.

As suggested by tom, taking the definite article is possible but not obligatory either, which also confirms that these are quasi-proper nouns.

Now, the third example (I'm not a politician. I'm a queen.) is different from the first two in that queen here is not used as a quasi-proper noun. Therefore, you can't capitalize the first letter (I'm not a politician. * I'm a Queen.). Nor can you leave out the article (I'm not a politician. * I'm queen.) In other words, the speaker here simply uses queen to describe her attribute as in politician rather than her identity.