There is a (textually) visible struggle going on in Holden regarding his future. Holden seems to want to escape the inevitable show-down with his parents. He will have to tell them that he got kicked out of Pency. This hesitance seems to color his view of all the future. The only positive view he has of the future comes in his dreams of escape to the west.
In the end, Holden is prepared to face his parents. That seems like growing up...a little bit.
Holden makes the connection of what he sees himself doing in the future when he plays baseball with his younger sister, Phoebe. She is perhaps the only person with whom he connects. From his observations, he concludes that children are the only people worth respecting. Therefore, the coming of age happens when he finally sees himself as a person who will eventually be of value: He will become a catcher in the rye, who will protect the children from "going astray" in life. This is certainly a significant thing for someone like Holden, who always viewed society as an enemy. Therefore, it can certainly count as a "coming of age" moment.
There is a definite sense in which this novel charts the way in which Holden reaches the lowest point of his life, having been kicked out of school and then going on some kind of journey by himself and discovering certain facts about himself and others. By the end of the novel, however, the scenes with Phoebe, his sister, show that he is capable of reaching out to others and that relationships are important to him. He in a sense grows up as he realises that he is not completely unattached and free to roam without consequences or responsibility.
In addition to what Pohnpei says, we also notice Holden letting go of the phoniness in his own character. Holden hates phonies, but we cannot help but notice his own character is perhaps the biggest phony through much of the novel. However, this nov el is told from the confines of the hospital where Holden has been trying to reclaim his health. So the very things he chooses to show us about himself are his efforts to highlight the phoniness he is overcoming.
To some extent, you can see Holden growing up in this book. He is moving (or at least trying to move) from being a self-absorbed kid to at least being someone who thinks about other people some of the time. We see him trying to understand the world around him. We see him trying to figure out who and what is phony in life and what is genuine.
I don't think we truly see Holden make a huge turnaround in his attitude, but I do think we see him struggling towards a change.