I bought the Imperial City shack. It started a new quest. I completed the
first part by visiting the shack. The second part of the quest is to buy
furnishings from Sergius Verus at the Three Brothers Trade Goods. I
bought the storage area, but the quest still directs me to Sergius Verus.
I tried visiting the shack but the quest doesn't update. Do I need to buy
everything for the house to advance/finish the quest?

Yes. Your next and final quest update on that will be after you have bought everything for the house.

Thanks.