By default QuickAdd is a plugin to help you add text to a note at one place. It doesn’t by itself duplicate stuff elsewhere.
If you add a task to a master list, or into your daily notes (like many do), you could use a query to showcase these tasks elsewhere. If that is what you’re looking for, you would need to look into either using the tasks plugin, or doing a TASK query with the dataview plugin. Either of these would allow for you to define your tasks in notes within your vault somewhere, and they could sort and filter and display these tasks according to your criteria (like tasks for this quarter).