I have trouble breathing and am always out of breath for no reason. Is this a symptom of fibromyalgia?

Possible. Get evaluated to get a proper diagnosis and treatment. Other causes of shortness of breath should be ruled out before assuming it is from fibromyalgia.
Many possibilities. Chronic shortness of breath (sob) usually stems from the lungs or the heart although depression, anemia and hormonal problems could cause fatigue and a feeling of shortness of breath. Fibromyalgia is not high on the list for feeling short of breath. Questions that may help: is your sob seasonal? Due to exercise? Is there swelling of your ankles? Do you wheeze? Cough? Did you smoke?