I have pain in left index finger joint. Will it go away?

Finger pain. It depends on what is causing the pain. Some causes of joint pain are self limiting but others require treatment. If the pain has been there for a long time and is not improving, you should see your doctor.
Most likely. Depending on the cause of pain, it is more or less likely to go away, but if it is pain only with no swelling or other joint involvement and no definite cause for it, it may be some muscular pain that warrants observation and simple measures (massage, heat/cold application, topical rub, tylenol/ibuprofen, finger splint in worst case scenario). If in doubt, see your doc.