How can I balance mandatory quests in two-player games of Lords of Waterdeep?
First, I agree that Mandatory Quests can be a little unbalancing. I've never played 2-player, but I can see how they could be even worse in that case. I've seen a handful of alternatives, any of which may work for you and your opponent:


Don't use them. Remove them from the Intrigue deck beforehand, or discard and draw again if you draw one during the game.
Allow, but penalise, completing other quests while they're in play. Make them not mandatory. Allow the receiver of the Mandatory Quest to complete other quests, but any VP reward is reduced by, for instance, the VP reward of the Mandatory Quest, until they complete the Mandatory Quest to remove this penalty.
Allow other quests, but penalise taking time over the Mandatory Quest. Allow other quests to be completed, but at the end of every one of your turns when you have an incomplete Mandatory Quest in front of you, lose some VP. You could lose 1 VP the first turn, 2 the second and so on, tracking with VP chips if necessary (or think of a flat rate per turn - 2 VP maybe? - if you prefer). You could probably ignore a Mandatory Quest late in the game (when they're most annoying, in my experience), but early Mandatory Quests should be completed before they snowball.
Allow counters. You can reveal a Mandatory Quest from your hand to cancel out being given one. Both are discarded. This would reduce how often they're played, since you may keep one in your hand in defence.


Hopefully one of these suggestions will help you out, or inspire you to come up with your own solution that you feel is right!