No, eating too many carrots will not harm your bearded dragon. Reaching a vitamin A overdose is pretty much impossible with plants as they will just pass it from their system.
You mention in your other thread that he eats the vegetables as soon as you put it in the cage. Does he eat them all? If he eats them all he is probably still hungry and should be given food until he stops eating. Bearded dragons know when they are full, over-eating is not a problem.