With food processor running, drop garlic through food chute; process until minced. Add anchovies, capers, and olive oil; process until finely minced. Add olives, processing just until chopped. (Do not overprocess; mixture should be chunky.)
Combine olive mixture and shrimp in a large bowl; toss well. Place 4 tortillas on a large baking sheet. Sprinkle each tortilla with 1/3 cup cheese; top with one-fourth of shrimp mixture. Top shrimp mixture with remaining cheese. Place remaining tortillas over cheese, pressing down gently.
Grill, covered, over medium heat (300º to 350º) for 2 minutes. Quickly and carefully turn quesadillas, using a large, wide spatula. Grill, covered, 2 additional minutes or until cheese melts and tortillas are golden. Cut each quesadilla into 6 wedges, and serve immediately.
Tip: Turning the quesadillas quickly keeps their juices from dripping, which can cause flare-ups and burned tortillas.