Of course He does. "God" is one of them. Others are - Lord, Yahweh, Elohim, Adonai, Allah, Brahma, Krishna, and more too numerous to mention.
I think what you mean is - Does God have a name He calls Himself and which He has revealed to humankind? Some say yes, some say no.
In any case, what if He does or He doesn't? What is the import of your question?
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