In a 4 1/2-quart or larger electric slow-cooker, mix the onion, bell pepper, garlic, jalapeño, chili powder, oregano, and cumin.
Pull off and discard skin from turkey thighs; trim off and discard fat. Rinse thighs and lay on vegetables in cooker. Pour hominy and tomatoes with juice over turkey.
Cover and cook until turkey pulls easily from the bone, 6 to 7 hours on low, 4 to 5 hours on high. If possible, turn meat over about halfway through cooking.
With a slotted spoon, transfer turkey to a plate. Skim and discard any fat from cooking liquid. When turkey is cool enough to handle, in about 10 minutes, discard bones and tear meat into large chunks. Return meat and juices to cooker; cover and cook until hot, 10 to 15 minutes.
Spoon turkey chili into wide bowls. Top portions as desired with cheese, olives, and green onions. Add salt and pepper to taste.