Would I be eligible for unemployment benefits if I quit my job due to postpartum depression?

No, unfortunately, you would not be. When someone voluntarily separates from employment (quits or resigns), even for what she believes to be good reasons (e.g. depression; a too-stressful job; etc.), that person is ineligible for unemployment compensation. Only when somone is fired, terminated, or laid off (and not fired "for cause") is she able to receive unemployment benefits.