Preheat oven to 425 degrees F (220 degrees C). Grease a large baking sheet with sides.
Place chicken quarters on prepared baking sheet. Stir potatoes, olive oil, juice of 1 lemon, basil, oregano, salt, pepper, and lemon herb seasoning together in a large bowl to coat potatoes. Arrange potatoes around chicken on baking sheet. Pour about 3/4 of oil mixture over chicken, reserving remaining oil; drizzle remaining lemon juice over chicken and potatoes.
Bake in the preheated oven for about 30 minutes; shake baking sheet to loosen potatoes, then continue baking for 15 minutes. Place green beans in reserved oil mixture; toss to coat. Remove chicken mixture from oven; pour green bean mixture over chicken and potatoes.
Return pan to the oven; bake until green beans are tender with a bite, chicken is no longer pink at the bone and juices run clear, about 15 minutes. An instant-read thermometer inserted near the bone should read 165 degrees F.