Because of their high content of enzymes and antioxidants, all vegetables are good for liver heath, but particularly carrots, kale, and broccoli. These provide vitamins that are needed for nutrition. Eating vegetables has significant benefits for liver function. The secret of vegetables is that they cleanse the liver and thus eliminate the build-up of toxins and fats.
It's the combination of fibre, minerals, vitamins and certain complex compounds that make vegetables so important. An additional vegetable to eat is beetroot. Studies on rats have returned findings that beetroot juice reduces oxidative damage and inflammation. Presumably beetroot itself does, but the studies used only juice.