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.

iOS/Android/Windows Phone:
Follow these steps to change your display name in QuizUp:
  1. Tap the settings option in the top left corner
  2. Under settings you'll find the edit profile option
  3. In “Edit Profile” tap your display name to edit it into whatever you want (well, almost whatever you want; no profanities please)
  4. Tap save in the top right corner

Web:

You can also easily access the settings from QuizUp.com - just click the triangle besides your display name on the top bar and select Settings. From there you can change your profile and update your display name.