Melt ½ tablespoon of butter in a large skillet over medium-high heat. Add the ham, bell peppers, and onion and sauté until the vegetables are tender, about 5 minutes.
Add the eggs to the skillet and season to taste with salt and pepper. Cook the eggs, stirring often, until they are set and no longer runny. Transfer the egg mixture to a bowl or plate and cover loosely with foil to keep warm.
Melt another ½ tablespoon of butter in the skillet over medium heat. Add one tortilla to the skillet and top with half the cheddar cheese, followed by the reserved egg mixture. Top the eggs with the remaining half of cheddar cheese and add a second tortilla on top.
Cook the quesadilla for about 2-4 minutes until the bottom is golden and crispy, then carefully flip the quesadilla over with a large spatula. Cook the other side for another 2-4 minutes until golden and crispy and the cheese inside has melted.
Transfer the quesadilla to a cutting board and cut into 4 or 8 wedges.