When looking at potential buildings I can make, some of them say they require level x and completing some quest. Unfortunately, I've surpassed some of the levels required for these buildings but I can not seem to get the quest to be offered by any of the resident of my town. Is there any way I can trigger certain residents to offer me new quests?
Playing the game and completing the quests you currently have will eventually unlock new buildings. Once you do have those buildings build them to unlock new characters, each having their own new quests to complete.