What are the conflicts in Paper Towns?
As the previous answer pointed out, a central internal conflict is the conflict surrounding Quentin's feelings for Margo. He has idolized her, and he believes that he is in love with her; however, over the course of the novel, Quentin finally sees who Margo really is. The result is that he learns he can't make a life with Margo.
But I can’t say anything because she kisses me again, and it’s in the moment that she kisses me that I know without question that we’re headed in different directions.
There are some external conflicts in the book, too. A fairly straightforward external conflict is the conflict between Margo and her ex-boyfriend. Margo isn't just hurt over Jase's actions. She actually seeks out and takes revenge on him.
She pulled open the Lexus’s driver-side door, sat down in the seat, and proceeded to attach The Club to Jase’s steering wheel. Then she softly closed the door to the Lexus. “Dumb bastard never locks that car,” she mumbled as she climbed back into the minivan.
A bit later in the book, she explains her actions.
"Because he’s been cheating on me for six weeks? Because he’s probably given me god-only-knows-what disease? Because he’s a disgusting idiot who will probably be rich and happy his whole life . . ."
I believe another external conflict surrounds Quentin. Internally, he does wrestle with his feelings for Margo; however, externally he struggles with actually finding Margo. A great deal of the book is about his quest to find Margo. The conflict is made tougher by the fact that the police and even Margo's parents don't seem too concerned about her. Their lack of concern means that they lack the willingness to help Quentin find her.
He took a few notes while I talked, but nothing very detailed. And something about telling him, and his scribbling in the notebook, and her parents being so lame — something about all of it made the possibility of her being lastingly missing well up in me for the first time.
He's left to solve the conflict without their help.
The central conflict of this novel is the journey that Quentin, the protagonist makes as he battles with his love and infatuation for Margo and comes to realise the difference between the Margo he loves and the real Margo who is actually shown to be a rather callous and unthinking individual, as shown by her night of pranks that she insists Quentin joins her on. In spite of Quentin's rather balanced life, apart from discovering a suicide victim with Margo, it is clear in the novel that Quentin's character is burdened by his youthful infatuation with Margo, and it is only when he embarks on his quest to "save" Margo that Quentin is forced to see that Margo is actually very different from how he imagined her to be:
The fundamental mistake I had always made—and that she had, in fairness, always led me to make—was this: Margo was not a miracle. She was not an adventure. She was not a fine and precious thing. She was a girl.
The conflict in this novel therefore is an internal one and centres on the perception of Quentin of his childhood love and the gap between appearances and reality. Thankfully, Quentin is able to see Margo for who she really is and therefore move on in life to a healthier position.