Using small-hole strainer, drain liquid from Pico de Gallo; discard liquid. Transfer leftover tomato mixture to medium bowl. Mix in black beans, cheese and cilantro until combined.
Divide black bean mixture evenly over half of each tortilla (about ½ cup each). Fold tortillas in half.
Heat large griddle or skillet over medium-high heat. Brush with oil. Place filled tortillas on griddle. Cook, carefully flipping once, until tortillas are golden brown and crisp and cheese filling melts, about 5 minutes. Cut quesadillas into wedges.