When Hitler took over Germany the government of Hungary tried to form and alliance with the Nazis. In September of 1938, after the German annexation of the Sudetenland area of Czechoslovakia, some of the land was given to Hungary to make relations between the two nations. Over the next three years, as Germany invaded and took control of countries around Hungary, Germany gave Hungary possession of some more land areas. Later in 1941, Hungary and Germany went to war against the Soviet Union. After they took over Stalingrad, Hungary tried to get out of the alliance. In March 1944, Germany invaded Hungary. Many anti-Jewish laws were passed. The Jews were required to wear the Star Of David. Between May 15 and July 9 of 1944, Jews were being deported to camps. Then between July and October of 1944, the Jews in the Budapest were living in safety. However, on October 15, Hungary's government said that they were going to make peace with the allies, the Germans toppled the government. Power was given to someone named Ferenc Szalasi, and antisemitic, Arrow Cross Party. They brought terror throughout the city. There were so many Jewish people killed in that time.
Knowing this information, it seems as though Hungary never was very safe. It says that it was safe for around four months, but that's it. In Maus, they were captured on the train ride to Hungary. I am assuming that this was during the time when Hungary was in control of Ferenc Szalasi.