During world war 1 and two, did any conflicts break out between nations that were not involved directly with WW1 and WW2? NON- axis or allies countries I mean.
Perhaps not exactly what you're looking for, but the Winter War between the Soviet Union and Finland is technically one such. It happened after World War 2 had started and involved two countries that were at that point uninvolved in the greater war.
Certainly the Soviet Union and arguably Finland (depending on how you figure the Continuation War into the grand scheme of the World War) would take part in World War 2, but for the time being it involved no other powers than themselves.
Of course, it's hard to say they had nothing to do with the War, but then it might be tough finding a war which was completely unconnected to something as all-encompassing as the Second World War.
Depending on how you define things, the conflict (not a full-scale war at this point, more of an insurgency) in Mandatory Palestine could qualify. The element of it between Jews and Arabs didn't really involve WW2 terribly closely, and the Jews vs British bit, while muted during the war, still existed even though the Jews were Allies-aligned. WW2 was important in the historical progression of things, but the fighting during the war didn't define who was fighting who.
During WW1, the USA invaded Haiti and the Dominican Republic (albeit with minimal resistance) as part of the "Banana Wars".
During WW2, Peru defeated Ecuador in a brief border war.
The Soviet Union declared war on China and 'took' Xinjiang (installed a pro Soviet Union governor over the land during the Warlord period of China). As well as the other Xinjiang wars.
The Abyssinian Crisis of 1935, of course (though you want no Axis countries).
Russian Revolution still had some White Armies going around.
Japan invading Manchuko (also doesn't really apply).
Chaco War (between Paraguay and Bolivia)
Unification of Saudi Arabia wars
Mexican Revolution (Cristero War)
Chinese cliques-Tibet wars
And the many wars following the collapse of the Ottomans.
Are all the wars I can think of. Though there were many other smaller conflicts and rebellions.
Edit: Agaa, you were talking about DURING the wars and not the Interwar period.
Thailand went to war with French Indochina, which was with the Vichy regime.
There was also the Chaco War in 1935 which was between Bolivia and Paraguay.