You need to open the appropriate modal, let me see if I can find an example.
I think this deals with the same thing:
Edit: if you want to use the content picker or media picker property editor in your custom property editor, I think @BishalTimalsina12 approach is the way to go. But if you want to do something custom for picking content or media, you can use the modals directly and handle the result yourself. It depends on what you want to do
