How to say "This person asked me, 'what's your name?'" in Korean

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Translated it's "누가 제 이름이 뭐냐고 물어봤어요." 
Literally meaning, "Someone asked me what my name was."
If you want a closer translation to your sentence, then it's "누가 저한테 물어봤어요, '이름이 뭐예요?'"
But the first one sounds more natural if you're talking to someone.
Hope that helps!

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이사람이 '너 이름이 뭐냐?'고 물어봤어요.

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그 사람이 저에게 이름이 뭐냐고 물어봤어요