No, Japan's problems aren't due to Nigeria's high birth rates, but many of Nigeria's problems - poverty and crime - are partly due to the fact that people in the slums are having many more children than they or their nation's economy/infrastructure can reasonably provide for. And many of the developed world's problems with mass immigration are due to high levels of immigration from more fertile Third World countries.
Developed countries do not have to admit mass quantities of foreigners. They are choosing to do so, in part because they do not have sufficient workers/taxpayers among the ethnic-native population.
Cause and effect, chain of events. One can project as much as one likes, but most countries have self-generated their own set of problems through the actions of their own citizens.