The New Sydney police bureau was the police force of the planet New Sydney.
After Miles O'Brien disappeared there in 2375, Yanas Tigan called in several favors from them. The bureau had heard of a Starfleet Intelligence operative being on the planet and later found O'Brien being beaten up by several Nausicaans. Lieutenant Fuchida later took him to the Tigan's home.
After it was revealed that Norvo Tigan was responsible for the murder of Morica Bilby, he was arrested by the Bureau. (DS9: "Prodigal Daughter")
The name of the police force was not mentioned in dialogue, but is taken from the script. It seems they also have jurisdiction over the wider Sappora system.