Not true. Amin al-Husseini supported Hitler from the get-go because the Arabs resented so much Jewish immigration to Palestine. Jews had gone from about 10% of the population in 1920 to almost one-third in just a couple of decades, which, of course, is how the conflict began.
I'd agree that it's not Hitler's fault, but he definitely fanned the flames and it's interesting how much lasting influence the Nazis have had on Islamic antisemitism.
Immigration goes back before that, the Ottoman Empire actually invited Jews to live in Palestine and gave them land.
During World War I, the British promised German Jews (Zionist movement) that they would give them land in Palestine to create Israel, their motive being to weaken their support of the German war effort. Note that Germany and the Ottomans who controlled Palestine were allies at the time. The British also promised Palestinian Arabs that they would allow them to create their own country if they defeated the Ottomans, and many Palestinian rebel groups joined the British because of this.
So the British promised Palestine to European Jews and to the Arab Palestinians during World War I, but didn't uphold either promise and instead kept the land for themselves.