Is it common in anxiety to have shortness of breath, chest pain and palpitations , anytime no matter if you are relax and keep having it for months?

NO: If you are having chest pain and difficulty breathing you should go to emergency department to be evaluated for serious disease. Do not dismiss these symptoms.

Yes but: Need to rule out other problems as well--i'd start with your family doc.

Certainly: Anxiety is associated with physical symptoms like the ones you mentioned, they can be severe enough that they land people in the er. Automatic thoughts (as we call them) can be working in the background behind the curtains in your mind leading to anxiety, this happens while you are are not consciously thinking of them . . Cbt explores and addresses these thoughts with you.See a professional.