Can I be disgnosed with fibromalygia even that I don't have the other symptoms but chronic back pain?

No. No, typically fibromyalgia requires certain elements in order to be diagnosed. All to often it becomes a diagnosis where you are labelled without clear signs or symptoms. Understandably this is a clinical entity that we don't understand all that well but having chronic low back pain is not typically by itself consistent with fibromyalgia.
Not likely. If your only symptoms is back pain, the cause of the back pain shoud be looked into with examination and tests like a mri. This will likey reveal the cause of your back pain. Fibromyalgia presenting with only back pain is unlikely.