Could it be possible to dislocate a finger joint without serious pain?

Anything is possible: I suppose this could be the case and depending on the laxity of your joints, any underlying issues you may have such as marfan's syndrome, or a pain tolerance condition such that you don't feel as much pain, but otherwise, you should have a very painful, swollen, deformed looking finger and i would recommend an examination.

Unlikely: If nerves are injured or someone has an inability to feel things i guess it is possible; however, for most patients the dislocation implies tearing ligaments that would be painful.