Your symptoms are somewhat concerning. I would recommend that you discuss this with your primary care doctor
. This should be evaluated.
In general, when exposed to cold, the body works to defend its normal temperature (98.6 degrees F). Therefore, the blood vessels in the skin constrict down so less blood is exposed to the cold air and less of your body's heat can diffuse out. This is to say, your fingers becoming a little white (as the blood flow to the skin diminishes) is somewhat normal.
There is a possible concerning cause of this known as Raynaud's phenomenon. Raynaud's is a pathological constriction of the blood vessels in response to cold. This normally involves all of the the fingers and often the toes. The phenomenon by itself is not a major condition, but unfortunately it is sometimes associated with other diseases. Therefore talking to your primary doctor to rule out these other disease is important (if it truly sounds like Raynaud's). There are autoimmune diseases (like lupus), cancers and other endocrine disorders that should be ruled out.
The other possibility is a narrowing of the blood vessel feeding the finger. If you are a smoker this can be known as Beurger's disease. I recommend that you see your doctor as soon as possible.