In Star Trek: The Next Generation, Captain Jean-Luc Picard has many awkward interactions with children and teens. In the two-part pilot episode of TNG, “Encounter at Farpoint: Parts 1 & 2,” Picard first meets Wesley Crusher. On the bridge, Captain Picard (Patrick Stewart) notices the turbolift doors open, sees Wesley (Wil Wheaton) before realizing that he’s accompanied by his mother, Beverly, and sternly says, “Children are not allowed on the bridge.”
Wesley Crusher becomes a recurring character after he is introduced in the pilot episode. Picard’s disdain for the teen comes to a head when he shouts, “Shut up, Wesley!” in season 1, episode 13 “Datalore,” though the insult was obviously uncalled for since Wesley was trying to warn the crew about something they couldn’t see. Picard has many more terse encounters with Wesley throughout the show, but Captain Picard doesn’t seem to care for any of the other children on board, either.
After Star Trek: The Original Series, children have appeared frequently in Star Trek series like Star Trek: Depp Space Nine and Star Trek: Voyager, but it really started with TNG. This is because the Enterprise-D is a new kind of galaxy-class vessel that has hundreds of civilians on board, including family members of the crew. Since the Enterprise is a new kind of Starfleet ship, everyone on board is forced to adapt to the radical idea of having civilians coexist with Starfleet officers in the confinement of a spaceship. It makes sense that Captain Picard, especially, would need some time to warm up to the idea since he is responsible for commanding the crew and protecting the passengers – something he didn’t necessarily have to worry about before on his previous mission.
Picard’s previous Starfleet assignment was on the USS Stargazer where he served as commanding officer for 22 years. Unlike the Enterprise, there wouldn’t have been children on the Stargazer (at least not as many), so Picard simply lacked experience interacting with children. Understandably, this made Picard uncomfortable with children and teens on the Enterprise in TNG. It was also on the Stargazer where Picard met Jack Crusher, Wesley Crusher’s father, who died on an away mission, leaving the widowed Beverly to care for her son. This explains why Picard initially allowed Wesley to explore the bridge, despite his rule prohibiting children.
Picard didn’t really begin to get over his discomfort around children until Star Trek: The Next Generation season 6, episode 7, “Rascals,” when Picard was turned into a child himself and began to understand the Enterprise from a child’s perspective. And by the time of Picard, the retired Starfleet officer seemed to get along well with Kestra, Deanna Troi and William Ryker's teenage daughter. While Picard’s irritable behavior toward children (particularly Wesley Crusher) is not always justified in Star Trek: The Next Generation, it's somewhat explained by the fact that he hadn’t spent time around children for two decades and that he felt intense pressure captaining the first vessel to transport children on a Starfleet mission.