How does Jay Gatsby represent integrity morality and honesty?

This is very much a matter of your own interpretation and there is room to argue that any character in the book is the most moral.

I would personally argue that it is part of the book's message that ultimately none of them are moral.

Nick Carraway starts off the book by boasting that he was taught right and wrong from a young age and that perhaps he is more moral and neutral in his judgments than others. However, as the books moves on we can see this as being a lie. Nick does judge people in the book and he often does this quite harshly. He also allows himself to fall in with a bad crowd and take part in the decadance of his age. Nick's quite naive about his faults though and so although he is immoral he's not aware of it.

Daisy Buchanan is immoral because she leads on Gatsby's hopes of a partnership and then drops him cold when her husband steps in. She is a flirt and is dishonest with Gatsby and her husband. She is also consumerist and allows herself to follow along with the decadances of her time. It's also worth remembering that Daisy was happy to drive off after she'd hit a woman with her car and is happy to let Gatsby take the blame and the fatal punishment. I would say she is the least moral.

Jay Gatsby is immoral because he tries to lead astray a married woman. He's also a criminal and a rum runner during prohibition. However, I would argue that Gatsby is the most innocent about his crimes and since he also sticks by Daisy, is in some ways the most moral of them all.

Jordan Baker is a liar and a cheat. She's also morally blurry with men.

Tom Buchanan is racist, closed minded and violent. He looks down on others and thinks he's better than everyone else. A little like Gatsby though, Tom is more innocent in his faults than some of the women.

To be honest though you could argue this either way.