Heat butter in a saucepan over medium heat; cook and stir onion until slightly tender, about 5 minutes. Add ground beef, Worcestershire sauce, salt, black pepper, and garlic powder; cook and stir until beef is browned and crumbly, about 10 minutes. Remove from heat, drain liquid from beef, and let sit for 2 minutes.
Heat a griddle or flat pan over medium heat and spray with cooking spray.
Place 1 tortilla on the hot griddle, top with some of the Colby cheese, 1/4 of the beef mixture, 1 teaspoon ketchup, 1 teaspoon mustard, and 1 teaspoon relish. Top with more Colby cheese and 1 tortilla. Flatten the quesadilla with your hand.
Flip quesadilla with a spatula and cook until browned and cheese is melted, 3 to 5 minutes more. Cut quesadilla into 4 pieces. Repeat with remaining tortillas, ground beef, cheese, ketchup, mustard, and relish.