Someone asked me "how is it going?"
I reply to her: "It's been going pretty good. Thanks for asking! How about you?"
Is there any grammar mistake in it or this sentence is alright?
Most people don't talk like that. To answer, "It's been going pretty good," is a bit stilted. More natural would be, "Okay," or "Not too bad," or "Just fine." Any of these can be followed with "You?" or "And you?" or "And with you?" So, something like, "Okay. Thanks for asking. You?" would carry the dialogue in more natural way.
'Well' may be more grammatically suitable than 'good' , and 'what about you?' could be more proper as well, although both these are quite all right and acceptable, I think.