If you give a sim the good trait, one of the social options under that trait is "Give Comforting Hug". The game says a sim must truly be suffering for this interaction to work, but it's almost always accepted, bad day or not. Awww.
Some of the quests count. There is one in particular where a child has disappeared and has to be found. When the monarch finds him, he says something along the lines of, "I ran away because I heard the monarch was vicious and cruel, but obviously, I was wrong. Thanks!"
In the "Heir to the Throne" quest, the Monarch has the option to choose whether or not they want a prince or princess. No matter what the gender is, the Monarch becomes immediately attached to his/her little baby and decides the prince/princess will be the heir to the throne.