In the sentence, "Neither the judge nor the witness believes him to be innocent (ജഡ്ജിയോ സാക്ഷിയോ അവൻ നിരപരാധിയാണെന്ന് വിശ്വസിക്കുന്നില്ല)," we have two subjects joined by the conjunction "nor": "the judge" and "the witness."
When two singular subjects are joined by "nor" or "or," the verb agrees with the closer subject, which in this case is "the witness." Therefore, we use the singular form of the verb, "believes."