Captain Picard Day was an annual event held aboard the USS Enterprise-D for the ship's schoolchildren. The day's activities included a contest, judged by Captain Jean-Luc Picard himself, in which the children created paintings and models of the captain. Picard wasn't very fond of the event himself, as he generally did not respond well to children.
In 2370, a seven-year-old named Paul Menegay won the contest with an orange clay sculpture of the captain's head. Much to Picard's dismay, Commander William T. Riker took great pleasure in the proceedings. As a means of revenge, Picard arranged to have a "Commander Riker Day" the following month, and planned to enter the competition himself. (TNG: "The Pegasus")
Despite his apparent disdain for the event, Picard retained a banner from the one such celebration within his vault at the Starfleet Museum Quantum Archives, where it was visible in 2399. (PIC: "Remembrance")
"They seem to have a rather exaggerated impression of me."
"I don't know; I think the resemblance is rather striking. Wouldn't you agree, Number One?"
- - Picard and Riker, while Riker holds a doll of Picard and imitates his voice
"Captain Picard Day?"
"Oh uh … Yes it's a … it's … it's for the children. I'm a … (nervous chuckle) I'm a role model."
"I'm sure you are. Starfleet out."
- - Admiral Margaret Blackwell and Picard, regarding the decorations behind the latter during communications
"You know, I'm sorry, but I'm starting to think that Barb might not actually exist."
"Oh, she's real. She's as real as a hopped-up Q on Captain Picard Day."