There are maybe three questions here, while I'll try to address in brief.

First, did Goering invoke Hiroshima in his defense at Nuremberg or elsewhere? Not to my knowledge. If he did, I haven't seen it, anyway.

Second, did Hiroshima come up at the Nuremberg trials? Yes. In several of the prosecutions, notably "The Hostage Case" and "The Doctors' Trial," Hiroshima was invoked as a way of implying that a truly fair trial would also either extend to the Allies as war criminals, or that the Allies were not hypocritical they would not consider the Nazi actions to be unique atrocities. It had no apparent effect at the trial, but it was part of an argument made by the defense. You can see some of these arguments here. But this is not quite the same thing as the quote from the movie, which is about the racial dimension to the use of the bombs (see below).

Third, when did people start implying that US used the atomic bombs against Japan for racial reasons, and might not have used them against Germany? I don't know exactly when this argument began cropping up but it has a definite post-1940s and even post-1950s feel to it. I suspect it is not something that gained much circulation prior to the 1970s, when these kinds of arguments about racial injustice became more "mainstream" in general. There are some cases of non-American, non-European voices asking this question in the 1940s (e.g. by mid-1945, an editorial for an Indian newspaper asked this question), and certainly by the 1960s this had become part of anti-racist, anti-colonial rhetoric (the Black Panthers, for example, including Hiroshima in a list of racial crimes). The most famous example of this argument comes from a post-Cold War book by Ronald Takaki (Hiroshima: Why America Dropped the Bomb, 1995), which was largely rejected by (predominantly white, male) scholars of the atomic bomb (arguably prematurely).

In any case, there have been many who have asked this question, sometimes seriously (e.g., as a serious intellectual engagement), but more often flippantly (e.g., as a cheap political point to be made). What is tricky is that the flippant approach is very easy to refute with an equally flippant response (e.g., the Allies did extensively bomb German cities as well, and the timing of the development of the bombs was such that it could not be considered for Germany, and in any case apparently Roosevelt did ask about using the atomic bomb against Germany with no apparent reservations) that is true but also ignores the obvious and persistent effects of racism in the US campaign against Japan (which are very easy to document, and all serious study of race and racism these days does not see it as a simple "variable" to be tweaked but as an integral part of the context of decision-making and warfighting). Much more could be said about this, but I think the key point here is a) there are people who have made this argument, b) it's more complicated than it might at first appear, and c) putting these words in Goering's mouth feels pretty ahistorical.