Heat butter in a skillet over medium heat; cook and stir onion and ham in butter until onion is translucent, about 5 minutes. Stir in morel mushrooms and cook, stirring often, until the mushrooms release their liquid, about 2 minutes. Sprinkle in black pepper and remove from heat.
Fit pie crust into a 9-inch pie dish and crimp the edges with a fork. Poke the bottom and sides of the pie crust several times with the fork.
Bake crust in the preheated oven just until lightly browned, about 10 minutes.
Whisk eggs, cream, and flour in a bowl until well blended; stir in Colby-Monterey Jack cheese.
Spread ham and mushroom mixture into the pie crust and pour egg mixture over the filling.
Bake quiche in the preheated oven until crust is browned and the filling is set and lightly browned, about 45 minutes. Serve hot or cold.