I'm making this guide as there are multiple similar questions around.
1) Make sure you meet all the requirements for the quest. Make sure you are at the correct level and have all the buildings and quests prior to this quest finished.
2) Make sure you have all open storyline quests finished. A lot of quests won't trigger until certain other quests are finished and closed.
3) Make sure the character that triggers the quest is free and isn't on a job. We have a full list of gameplay articles here. Click on the article you want, sorted by levels, events etc. and then find the quest you want. Each quest part should say something along the lines of "After tapping on <Character>'s exclamation mark" where <Character> is the name of the character that triggers the quest.
4) By now, you should have gotten the quest. However, there is still a possibility that you haven't. If you haven't, try storing the character that triggers (unless it is Homer or Lisa as they can't be stored). Click on the character or building associate with the character and store it. After bringing them out again, they should trigger the quest.
There is still a possibility that this doesn't solve the issue and the quest doesn't trigger. You have to then do things outside of the game to try and trigger it.
1) Force close your app and reopen it. Get the list of apps open up on your device and shut it down from there. If you don't know how to do this, look up how to force close an app on your device. Upon reopening it, it should trigger.
2) If that still didn't work, try resetting your device. Turn it off then on again. Then open Tapped Out and look again to see if it triggers.
3) The final thing you can try is to uninstall the game and reinstall it.
4) In the slight chance after doing all these steps, the quest still doesn't trigger, either you have not met all the requirements for the quest or there is a problem with your game. Make doubly sure you meet the requirements and if you do, contact EA for help.