Position a rack in the center of the oven; heat the oven to 200°F.
In a small pot, bring 2 cups water to a boil over high heat. Add the garlic cloves and cook until soft, about 5 minutes. Drain the garlic, let it cool slightly, peel, and put it in a small bowl. With a fork, mash the garlic to a coarse paste and then add the butter and mix well.
In a medium bowl, combine the Monterey Jack, Parmigiano, and goat cheese. Spread the garlic butter on one side of each tortilla and set the tortillas on a work surface, buttered side down. Distribute the cheese mixture among the tortillas, covering only half of each and leaving a 1-inch margin at the edge. Fold the tortillas in half to enclose the filling, creating a half-moon.
In a 10- or 12-inch nonstick skillet over medium heat, cook two of the quesadillas, covered, until golden brown, about 4 minutes. Uncover and flip the quesadillas. Cook until the second side is golden brown and the cheese has melted completely, about 2 minutes. Transfer the quesadillas to the oven to keep warm and repeat with the remaining two quesadillas. (You can hold the cooked quesadillas in the warm oven for up to 30 minutes.) Cut the quesadillas in half (or smaller wedges if serving as an hors d’oeuvre) and serve.