In the first film, Frollo uses the place of justice to bring people like criminals or smuggled gypsies to be tortured and whipped. Captain Phoebus is introduced right when he's searching for the Palace, where he meets the judge for the first time. In the second film, Phoebus is seen using the Palace of Justice as more of a prison, as show when Madellaine was brought here after she was arrested.
Despite its name when Frollo was in control, it was not a place of justice but rather a place of fear, injustice, and tyranny, as Frollo rounded up gypsies and had them tortured to death in this dark place. After Frollo's death and his soldiers surrendered to the French army, it is under control to Frollo's successor who was not shown and serves Phoebus now as the Captain of the Guards by the King of France who also leads the French army against Frollo and his soldiers. It also has a jail cell.
The Palais de Justice located in the Île de la Cité in central Paris, France, is built on the site of the former royal palace of Saint Louis, of which the Sainte Chapelle remains. Thus the justice of the state has been dispensed at this site since medieval times. From the sixteenth century to the French Revolution this was the seat of the Parlement de Paris.