Make the dipping sauce by combining mayonnaise, dill relish, ketchup, mustard, garlic powder, onion powder, and vinegar together. Place in the refrigerator while you make the quesadillas.
For the quesadillas, brown the ground beef in a skillet along with the onion. Cook until beef is fully cooked and drain off any excess grease.
In a skillet, melt 1/2 Tbsp of butter. Place 1 tortilla in the pan. Top with a layer of 1/4 of the cheese followed with a layer of 1/4 of the ground beef. Then top with another tortilla (buttered side out).
Cook for a few minutes until the one side is browned and then flip, cooking until that side is browned and the cheese is melted. Repeat the same process with the remaining ingredients.
Cut each quesadilla into 4 wedges with a pizza cutter and serve with the dipping sauces. Serves 4.