Prepare grill to medium-high heat.
Combine first 4 ingredients in a small saucepan. Bring to a boil, and cook until reduced to 1/4 cup (about 8 minutes). Strain mixture through a sieve over a bowl, reserving wine mixture. Discard solids. Add wine mixture and jam to pan; cook over medium heat 2 minutes, stirring occasionally. Remove from heat; stir in butter. Keep warm.
Sprinkle the quail with 1/4 teaspoon kosher salt and ground black pepper. Place quail on a grill rack coated with cooking spray; grill 2 minutes on each side or until desired degree of doneness. Serve with wine sauce.