"Our son has been born! We are very happy! Thanks to all for your messages," the Barcelona FC player, 25, wrote in Spanish.
The timing of the announcement has prompted speculation that Pique has pulled a fast one, since Shakira, 35, has not confirmed the news.
Pique later seemed to be letting fans know it was a joke by tweeting, "He already has a name! He is called Innocent! Happy Day of the Innocents to all!"
Shakira revealed her pregnancy last September when announcing the cancellation of a Las Vegas gig.
The singer began dating sportsman Pique in 2011 after splitting from her longtime boyfriend Antonio de la Rua.