I am not sure what is going on here. Does the finger also appear blue or white? Is there pain? Is there numbness
or tingling? Or does it just feel cold? Have you injured it in the past? The answers to these questions might point us in the right direction.
For example, if you have broken or otherwise injured the finger in the past, it is possible that you will permanently have some changes in blood supply or nerve supply that might be the cause of what you are feeling.
Similarly, if you also have numbness, tingling, or weakness this might be a sign of a nerve problem. The ulnar nerve supplies this part of the hand and it can sometimes become pinched or inflamed (think about what happens when you hit your 'funny bone')
If the finger turns blue or white when exposed to cold temperatures, then this may be a variant of Raynaud's phenomenon, which is a form of circulatory problem.
I would suggest that you start by going to see your primary care doctor
. They will be able to examine the hand and determine what could be going on and whether any additional workup or treatment is needed.