If your name is Maddie Green, you should introduce yourself as マディ みどり(緑), or not.
Is this your question?
I recommend you not to do that.
Just say マディ グリーン as it is.
And later, if you mention your surname and say like "じつは、わたしのなまえは、マディ みどりです。",
that might be nice to break the ice!