Must have talked with Josephine about Antiva. It is not possible to get this quest before. The Inquisitor must have romanced Josephine, naturally, and done the Heraldry from a Herald quest.
When the player walks into Josephine's office, a cutscene starts. She tells the Inquisitor that her parents --being unaware of the romantic situation between you -- have arranged for her to be engaged to a nobleman from her childhood and that she cannot engage in romantic activities until this is resolved. After the cutscene, the next time you speak with her, she will mention a way out of the engagement: if the Inquisitor challenges Otranto to a duel and wins. Josephine objects to the idea, fearing for your safety. After this dialogue, the war table mission Challenge Josephine's Fiancé to a Duel will become available. Once completed, a messenger appears in Skyhold's courtyard to inform the Inquisitor that Otranto is happy to duel in Val Royeaux.
While speaking with Josephine, a second war table mission can be triggered: Gather Information on Lord Otranto. This mission is optional, but completing it will unlock an additional dialogue option during the fight.
The Inquisitor may now travel to Val Royeaux to duel Otranto in a cutscene. Shortly, Josephine interrupts the duel, angry that the Inquisitor would risk their life. The player can now choose between declaring their love for Josephine, or allowing Lord Otranto to marry her.
If you opt to continue the romance, Josephine kisses the Inquisitor. Lord Otranto accepts defeat, having thought that they were simply having "an affair of passion," but admitting he cannot stand in the way of true love. The Inquisitor and Josephine return to Skyhold, and a cutscene will play as they cuddle by the fire.
Should you instead decide to break off the romance, Josephine refuses to marry either of her suitors and storms away in disgust. Otranto suggests declaring the duel a draw, at which point the Inquisitor can either agree or decide to continue the fight.
Officially in a relationship with Josephine.