Are both sentences OK?
I thought the question was asked me.
I thought the question was directed at me.

Question

Answer

Yes but the first sentence needs another word otherwise it is incomplete.

Answer

The first sentence needs one more word: "I thought the question was asked of me."
The second sentence is correct.

(One other possible variation that, while falling out of use, has the same meaning is "I thought the question was put to me.")