This question is prompted by a question on English Language and Usage: Can “mongrel” be used to refer to people? The OP of that ELU question is translating a fantasy novel into English.
My question is not whether this word is appropriate in the OP's (or any context) now (or ever), but whether, in fact, Mr. Spock was ever called a mongrel in a Star Trek TV episode or movie, and if so by whom and in what context.
I did find an interview with Leonard Nimoy in which Nimoy referred to Spock as a mongrel. See Nimoy Interview in which Nimoy says:
What kind of person should I be? And Spock seemed to have a fix on that. He had found a way to deal with those questions. In spite of the fact that he was not really acceptable amongst humans totally because he was strange, he was not totally acceptable among Vulcans because he was part human, he was a mongrel.
Kirk called Spock a half-breed in What Are Little Girls Made Of, but Kirk was not in his right mind when he did so.