"Who is looking for a judge?"

Translation: Ki keres bírót?

[MethodGT]

Why can't it be "Ki egy bírót keres?"

[Hulk014]

That would be "Ki keres (egy) bírót?". But in some cases "a" can be leave out in Hungarian.

Note: Verbs always follow interrogratives ( What - Mi?, Who-Ki/Kit?, Where-Hol? ect )

[ErikAnderson3]

Is ki bírót keres incorrect? If so, how?

[RyagonIV]

Might help if you read earlier comments here. :)
But I'll take this opportunity to elaborate a bit. If you have a question word, you want to put the focus of the sentence on that. You're interested in who is looking for the judge. In Hungarian, focus is determined by the verb: the things that are directly in front of the verb are the focal point in your sentence. So ki and keres have to stick together.
You are left with two possible structures here: "Ki keres (egy) bírót?" and "(Egy) bírót ki keres?"

[ErikAnderson3]

By extension from your explanation, then, if one is asking "who is looking for a judge (and not an eggplant or an elephant)?", with the emphasis of the question on the object of the verb, then the bírót would come just before the verb, yes?

[RyagonIV]

That.. might stretch it a bit. I have trouble thinking of a situation where you might ask that question without really wanting to know who is looking. If you want to put more emphasis on the judge, you say "Bírót ki keres?" (nem pedig padlizsánt vagy elefántot).

[ErikAnderson3]

Thank you, that's a very informative and helpful explanation. :)