You will have to decide for yourself whether or not you agree with this statement, but I do not think the novel reflects this. If we look at all the characters, setting aside Nick, about whom we don't know a great deal in terms of happiness or unhappiness, there is really only one character for whom this might be true. For those who do have wealth, there is no evidence to show that they dreamed that wealth would make them happy or that it did so.
Let's begin with the Buchanans. The Buchanans are already wealthy, with what appears to be family money. So any dreams that Tom or Daisy have are not about having more wealth. Tom's "dream" of happiness seems to center on having another woman over whom he can wield power, another person whom he can impress, since Daisy clearly sees through him. Daisy's dreams might be to have a husband who does not have a mistress, or at least one that does not publicly embarrass her, but her choice of Tom over Gatsby clearly has nothing to do with money in and of itself, since Gatsby has is far wealthier than Tom. It is Gatsby's class that makes Daisy reject him, in spite of his having more money.
Gatsby's dream is of Daisy, not of wealth. All of his wealth has been acquired as a means of winning Daisy. All of his efforts are meant to impress her, from his parties to the stories he tells about himself. His money has not brought him happiness, except to the degree it has brought him closer to Daisy. There is no temporary happiness for Gatsby, and he dies having only managed to get closer to Daisy, but not to win her.
George Wilson may have some dream of wealth, but he never achieves it in the story, so he provides no evidence for this thesis. He is a relatively poor, hard-working man, whose dream, finally, is of getting his wife back.
It is only Myrtle Wilson who might bear out this idea. It seems unlikely to me that she would commit adultery with a poorer man. She loves Tom's wealth. Buying a dog on a whim, having a nice apartment in the city, and flaunting new dresses before her sister all seem to bring her pleasure. So, it could be reasonably said that she exemplifies the theme expressed in the question.
But all in all, as I have read the book (many times), I do not see this as Fitzgerald's message at all about the American Dream. And if you do a little research on his life, you will find that he surely learned that money does not buy happiness.