Nihilism is not the only reason people don’t have children. For some, it is a deep concern for the world, the very opposite of nihilism.
No, but it (nihilsm) seems a common thread in present-day European culture, with people living pretty much for themselves. It could also be argued, that if “concern for the world” includes the premise that man is just another animal, a mere accident, then that could be unpacked to nihilism as well since it undermines any ultimate purpose such as that provided by the Judeo-Christian and other monotheistic worldviews.