Nobility is not a concept that comes to mind when I read The Great Gatsby. If nobility applies to any of the characters in this book, however, I would think it applies to Gatsby.
Gatsby has given up quite a bit to achieve his position and he is willing to give up even more to get Daisy. The risks he is willing to take might suggest some nobility of courage.
There seems to be much more to say about how Gatsby and the rest of the bunch have eschewed dignity in favor of glamour and ease. Tom's girlfriend is stepping out on her husband. Tom is stepping out on his wife. Daisy is stepping out on Tom. Gatsby is stealing one man's wife to take as his own.
Maybe if we look at Gatsby's quest as being one of self-definition we can say he possesses some nobility. He is unwilling to be defined by failure and by his past. He is determined to be a certain thing. There is some honor in finding a way to be true to yourself, even if that means making up an identity and lying to everyone you meet.
I would agree that the means of achieving his quest are not noble, but his actual quest -- the one we learn in chapter 9 that he has had since childhood,is actually very noble. When Mr. Gatz shows Nick the list of "goals" or "perfections" that Gatsby wanted to achieve, we learn that Gatsby's original vision of himself didn't rest on Daisy, but on achieving some semblance of what we commonly call the American Dream. Loving Daisy became a distraction to his once more noble calling. Remember, for example, that he left St. Olaf because he didn't think the college "could hear the drums of his destiny." That destiny was for wealth and happiness.
I would agree that his quest is not noble. He is pursing a woman who he thinks he is in love with but she is married to another man. It cannot be noble to try to take another man's wife. We must also consider the woman that Daisy is compared with the woman that Gatsby thinks she is. He is in love with someone that doesn't really exist. He has built up Daisy in his mind as the perfect woman, but she isn't really the person he wishes she was.
His methods to achieving his goals are certainly not noble either. He works with criminals and makes money from illegal activities. There's nothing noble in that.
I think his quest is not noble. You could say it is because he's pursuing true love. But really he's not. He's pursuing a woman who cares only about money. That doesn't seem very noble or very intelligent. I don't think there's anything noble about pursuing someone so shallow.