You just say .. She had a boy/ girl and its understood she gave birth to a baby. To say she gave birth to sounds odd actually. (American english). You could also say . she was blessed with If you're maybe a little religious . animals have litters . they also dont have births. But you will hear the term birthing more often in a veterinairy clinic than in a maternity ward.
I think the English translation should be, She had a little boy yesterday.
Having\Had a child means to be or become a parent. It just too long to use Gave Birth to.
English has no Specific Verb for describing Woman giving birth.
Some laungages do have, for instance Hebrew, ''Laledet'' a Single verb for describing this action.