Did Hera and Zeus have children together?

Yes: Area, Eris, Hebe, Eileithyia and Hephaestus.
Hera definitely had three kids with Zeus: Ares (Mars) (war god), Hebe (Juventas) (goddess of youth), and Eileithyia (Lucina) (goddess of birth-pain). According to different myths Hera had Hephaestus (god of the forge) by herself, or with Zeus.