Use a knife to cut the orange into 8 even pieces, keeping the peel on.
Place the orange pieces and the Granny Smith apple slices into a food processor or blender. Pulse 4 to 5 times to begin the chopping process.
Add the fresh cranberries, granulated white sugar, dark brown sugar, and lime zest into the food processor. Pulse about 8 to 10 times until it looks finely chopped.
Scoop into a large dish or bowl that can handle about 4 cups of relish. Cover and refrigerate for an hour to overnight to allow the flavors to meld.
Best when served cold.