This is the classic description of what we technically call poststatic dyskinesia or first step pain. That is pain that is worse after rest. There are multiple theories as to exactly what the physiological process is, however by far the most common reason for it is plantar fasciitis.
If the pain is a problem then plantar fasciitis is relatively straight forward to treat. As yours does not appear to be much of a problem during the day, then I would suggest that you just start with lots of calf muscles stretches. If that does not help, then see a podiatrist.