Although kept under wraps during the war for morale purposes, there were problems between Australian and American servicemen on occasions in Australia during WWII. There were numerous fights, usually alcohol-fuelled. The most serious was the so-called Battle of Brisbane which left one Australian dead and several with gunshot wounds, and many others injured on both sides.
http://home.st.net.au/~dunn/ozatwar/bob.htm
http://www.diggerhistory.info/pages-battles/ww2/battle-brisbane.htm
http://www.anzacday.org.au/history/ww2/homefront/overview.html about halfway down page.
There were plenty of smaller fights. My father got into one with US Marines in Melbourne’s best hotel, and got himself locked up by the cops while the Americans were left to drink in peace without being troubled by uncouth colonials. At the same time, my father had some friends among American troops at a major American camp near his home while his parents, like many other Australian families, often had American troops home for meals and company. This was a tradition that carried on for some years as I remember my father having visiting American sailors home for meals with the family in the 1950’s. (I don’t remember him feeding any Marines. )
It’s also worth remembering that many servicemen will happily fight anyone who isn’t part of their group. Just being from another regiment is enough, never mind another country. Add in grog and disputes over women and there’s bound to be trouble.
The problem expressed in the fights was more of a conflict between (some) servicemen from each country rather than expressing a general resentment by Australians toward Americans, as Australians generally were bloody grateful to the Americans for being here to fight the Japanese who threatened to invade us.