One major indication that Nick has grown during the novel is to compare his views on his age: when the novel starts, he is almost thirty years old, and view the upcoming decade with fear and disgust. Later, he takes refuge in his growing love for Jordan Baker:
Thirty -- the promise of a decade of loneliness, a thinning list of single men to know, a thinning brief-case of enthusiasm, thinning hair. But there was Jordan beside me, who, unlike Daisy, was too wise ever to carry well-forgotten dreams from age to age.
Nick tries to think of himself as completed by Jordan, who is in some ways more mature than he is. However, as the climactic events of the novel prove to him, she is less mature in the ways that matter to him:
"I'm thirty," I said. "I'm five years too old to lie to myself and call it honor."
She didn’t answer. Angry, and half in love with her, and tremendously sorry, I turned away.
(Fitzgerald, The Great Gatsby, mrbye.com)
Nicks realization is that he can't live the sort of life that the East Egg residents live, coasting on their inherited money and finding shallow diversions to keep them occupied. His exposure to Gatsby's great passions has changed him in ways that the superficial Jordan cannot understand; she believes that the events were just another scandal to be shared as gossip, but Nick can't get past what he experienced, and that critical divide shows how he has matured. Nick knows that Jordan will lie to herself and pretend that people and important events don't matter; Nick also knows that lying to one's self is far worse than lying to others, because the self always knows that the lie exists.
In the book nick tells very specifically about his fear of aging. He’s already thirty and he doesn’t like the idea of it at all. He says ‘Thirty – the promise of a decade of loneliness, a thinning list of single men to know...’ He realizes he still doesn’t have a wife and all the people he knows are getting married or have relationships. He’s afraid he will end up alone.
"Coming of age" means transitioning from child to adult-it's a porcess of maturation. Think about times in the book where you personally noticed Nick becoming more mature.