Use garlic and salt to make a paste. In a small bowl, combine with 1/4 teaspoon pepper and the oil. Rub all over pork.
Heat grill to high; oil grates. Place pork on hottest part of grill; cover. Cook, turning occasionally, until an instant-read thermometer inserted in thickest part registers 155 degrees. 10 to 20 minutes. Let rest 5 minutes; thinly slice across the grain.
Meanwhile, place onion slices on cooler part of grill; cover. Cook, turning once, until soft and beginning to brown, about 10 minutes. Reduce grill to medium-low.
Spread 1 tablespoon mustard on each tortilla, leaving a 1/2-inch border. Dividing evenly, layer half of each tortilla with ham, pork, onion, pickles, and cheese. Fold tortillas; press to close.
Lightly oil grates. Place quesadillas on grill; cover. Cook, turning once, until browned in spots and cheese has melted, about 4 minutes. Cut into wedges; serve immediately.