In a medium bowl, stir together the beans, peach, scallions, cilantro, lime juice, and salt.
Brush two of the tortillas with 1 1/2 teaspoons of the oil, then flip over and sprinkle entire surface with the goat cheese. Top evenly with about three-quarters of the bean-peach salsa, then all of the Monterey Jack cheese. Firmly press remaining tortillas on top. Brush with the remaining 1 1/2 teaspoons oil.
Fully preheat a grill or grill-pan. Grill the quesadillas over medium-high heat until the cheese melts and rich grill marks form, about 3 minutes per side.
Cut the quesadillas into 4 wedges each. Arrange on a platter, top with the remaining bean-peach salsa, and serve.