I’m neither Mormon nor Japanese, so no. But I have visited Utah and Japan, and know quite a few people from both cultures. They share many common characteristics. (As someone from a different part of Asia, Mormons are the most relatable white people to me in terms of the relationship between individuals and society.)
No, not anymore at least. The church used to in its early days (mid 19th century) encourage immigration to Utah (aka "Zion"), but many years ago they declared that "gathering of Zion" should take place where people already are.
Modern-day missionaries try to build up the local church where they are. Some people do end up wanting to emigrate to Utah, but this is not something the Church encourages.