It's an indirect relationship, as the Depression did not force us into war, and in fact, during most of the 1930s made us more isolationist. But as war in Europe heated up in the latter part of the decade, it became clear that the Allies, Britain and France, could not stand on their own.
FDR knew he had to get us in the war as soon as possible, but politically it might have gotten him defeated in the 1940 election, so he had to be careful. The Great Depression let him hide behind the idea of "job creation" when he passed the first peacetime draft. He also passed Cash and Carry and Lend-Lease programs so that America could sell war material to its friends while providing even more jobs to Americans. So if the Depression had any affect on the war effort, it was the fact that it led us down the path towards war and war production.
The great depression, which started in 1929, occurred much after World War I (1915 to 1919), and ended before World War II (1939 to 1945) started. There is no question of the great depression of contributing to war effort.
On the contrary some experts are of the view that increase in industrial activity in anticipation and preparation of World War II aided in speedier recovery from impact of great depression.