If you speak in front of a Japanese person and using Japanese, then basically it makes no sense to emphasize the word "Japanese" = 日本語, isn't it? :) Also, for formal situations, "you" should be given as 貴方(あなた) rather than あんた. So for this case, I would use:
貴方(あなた)の お名前は 何ですか？; or
[N.B. The prefix お (sometimes ご, with kanji = 御) in front of a noun indicates a respect towards the addressee.]
On the other hand, if you speak to other foreigners who are also learning Japanese, then you may perhaps add the phrase 日本語(にほんご), and the given sample in your question is in general correct.
P.S. [as to refer to the question supplement]
Okay, to that extent, I would say that it is fine to add the phrase 日本語の before お名前 if you insist on knowing the Japanese name of the addressee. However, this unfortunately shows a sort of strange phenomenon: some people prefer to use a Christian name rather than their own name. For instance, if you were in Hong Kong (with a majority of Chinese) and you asked someone here for their name, you would hear them introduce themselves using a Christian name far more than their own Chinese name!