Add quinoa and chicken broth to a small saucepan then bring to a boil over high heat. Place a lid on top, turn heat to low, then cook until quinoa is tender, 15 minutes. Fluff with a fork then set aside to cool slightly.
Meanwhile, season both sides of the chicken breasts with salt and pepper then grill over medium-high heat for 4-5 minutes a side, or until no longer pink in the center. (Alternatively you could saute or poach the chicken, or use leftover or rotisserie chicken.) Add corn cobs to the grill at the same time then rotate every couple of minutes to evenly char on all sides. Remove chicken and corn to a plate then let cool slightly. Shred or cut chicken into cubes, then mix with enough BBQ sauce to coat. Slice kernels off the cob then set aside.
Mix together Greek yogurt and BBQ sauce in a bowl. Evenly divide the cooked quinoa, BBQ chicken, grilled corn, black beans, and shredded cheese between four plates. Drizzle with yogurt and BBQ sauce mixture, then garnish with chopped cilantro.