Is it possible to complete a Quest card from the Quest Deck while there are still location cards in the staging area? Can you just ignore the location cards by not traveling to them and complete the quest?

Yes, you can. Traveling to a location is completely optional.

Keep in mind, the benefits of traveling to locations is solely that it reduces the amount of threat in the staging area. If you are capable of completing the quest without traveling to the location, it is to your benefit (usually, not always) to ignore it. In the same manner, the benefit of engaging enemies is that their threat is no longer in the staging area.

Thank You! This helps a lot