A small amount of grated carrot mixed in with a large amount of weeds or greens once in a while will not hurt anything. I wouldn't feed carrots often or in large amounts.
Russians cannot digest sugars properly - they cause digestive and kidney problems - fruit, tomato, bell pepper and carrot should only be fed very sparingly and very occasionally.
Personally I avoid it completely. I stick to leafy weedy greens and in winter supplement with a good quality pellet feed.