No, it did not. By invading Poland, Germany did not start the Second World War (as it is erroneously believed). It only started a local war between two neighboring countries that Germany quickly won.
World War II began not on September 1st, 1939 when German Panzers crosses the Polish border, but two days later – on September 3rd, 1939, when Great Britain and France declared war on Germany.
Both countries were global colonial powers and the British king was a nominal head of state in Canada, Australia and New Zealand. Which almost automatically brought these nations and dozens of colonies into the war. Thus transforming a local European conflict into a global war and starting World War II.
So, contrary to the most universal misconception, World War II was started not by Germany on September 1st, 1939, but by Great Britain and France on September 3rd, 1939. Sure, by invading Poland, Nazi Germany provoked World War II, but it did not start it.