Yes, bearded dragons can eat apple. In order for your pet to eat it, though, the apple has to be cut into bite sized pieces. Your lizard might like it a bit better if you peel the apple as well.
Yes, bearded dragons can eat bananas, but it is unhealthy. Bananas are high in phosporus, which binds with calcium, which can cause serious health issues such as Metabolic Bone Disease. Though your dragon can eat bananas, you should feed them bananas rarely. Best to stay away from this fruit altogether.