There are at least two cases mentioned in Arnold Kramer's "Nazi Prisoners of War in America":
One in which two Afrika Korps corporals held in a Colorado POW camp escaped with the aid of three Japanese American Nisei sisters who had been interned in another camp nearby. There is no evidence of the exact nature of the relationship, except that photos were taken of them embracing. A Kansas City Star article is the original source. Apparently the Germans also testified against the sisters at their trial for treason.
a 45 year old Mother of three GIs fell in love with a 22 year old German prisoner who escaped in September of 1945 with her aid and the two lived for a time as a couple in her mother's house. These facts also came to light at a trial after the pair were captured by the authorities.
It is of course possible that there were other instances when such cases did not result in a trial or publicity. An Army representative in Kansas cautioned farmwives against baking, sewing or giving rides to German POWs, which suggests a level of fraternization. (source here again is the Kansas City Star)