If traveling from Okinawa, Japan to Seoul, Korea, is it required to have a consent letter to leave and enter if the child is travelling only with the mother?
The Japan Ministry of Foreign Affairs notes that
A consent letter is not required when leaving / entering / departing Japan.
Similarly, the Republic of Korea does not make it a requirement. However, it recently became party to the Hague Convention on International Child Abduction and put in place procedures at entry/exit points. If one parent requests the ROK Immigration Service to place a travel restriction on his or her child, it may prevent the child from leaving the country, even if traveling with the other parent.
Having evidence of relationship and permission for the child’s travel from the parent not present, although not required, may avoid issues.