Preheat the broiler.
Clean out the quails, removing all the gizzards. Divide the rosemary into 4 equal parts and pat 1 part rosemary into the inner cavity of each bird. Slice the onions lengthwise so that each onion opens into large leaves. If using wooden skewers, soak them in warm water for 30 minutes before skewering the ingredients. Using 1 skewer per bird, skewer an onion leaf, 1 bay leaf, and 1 quail, taking care to skewer the bird directly through its inner cavity. Add a second bay leaf and a second onion leaf. Repeat with the remaining skewers. Place the 4 quail skewers side by side in a well-greased baking pan. Sprinkle the birds with salt and pepper and drizzle with the 1/4 cup of olive oil. Place birds on lower rack and broil about 10 minutes or until legs are well done and the rest is a medium doneness.