No, quite the other way around: the wealthier a country gets, the less kids are born as parents don't need as many of them to reproduce successfully. In fact this trend is so strong that countries like Japan or Germany are facing a really massive demographic problem as the working population shrinks over the next decades.
IMHO the reason is that in rich societies people has many possible activities to perform. They feel that parenting kids is not one of the most attractive ways to spend their time on. Furthermore raising kids cost more in rich countries than in poor ones, so they can't afford as many as their parents or grandparents could.
