You can customise the way you receive notifications.
When you've opened the QuizUp page (QuizUp tab), located in the middle of the tab bar you should see the Settings option located in the left corner on the top bar.
Tap the settings option in the top left corner
Under settings you'll find the Push notifications option
There you can enable/disable notifications. You can choose which Push Notifications you'll receive. Make sure the notifications are turned ON if you want to receive them and OFF if you want to stop receiving them
You can get push notifications when these things happen: you receive a challenge, a chat, a follow, follow request or acceptance, when someone likes or comments on something you've posted.
Doing this from the website, www.quizup.com, is very easy. Make sure that you are logged in at www.quizup.com and just click the triangle icon besides your display name in the top red bar and select Settings.