I think you can say
for "I'm on paternal leave" (In Japan, normally starting 8 weeks after the birth of the child and ending by the time when the child turns one year old... I think), and
for "I'm on maternity leave" (In Japan, normally for 6 weeks before and 8 weeks after the birth of the child... I think).
育児休業 is the right word.
If you are the woman giving birth, you could also use 産休.