Of Somewhat Fallen Fortune
This quest is available upon arriving at Skyhold if you take the time to talk to Josephine often especially about her family. Eventually she will confide in you about her families troubles and ask for your assistance. Speak with her enough and you will activate this quest and she will ask you to meet her in Val Royeaux. There you will encounter the person responsible for causing Josephine's troubles.
Once your meeting with Josephine is over in Val Royeaux Leliana will get involved upon your return to Skyhold. From there on you have the choice to allow Leliana to intervene in the situation to speed up the quest by doing one operation or you can choose to do what Josephine suggests which involves several operations and a few more trips to Val Royeaux.
GAMER NOTE: No matter how you choose to approach the situation Josephine will be appreciative and ask to meet you again in Val Royeaux. You will have an option to continue romancing her.