The example is somewhat awkward, but I think I know what you're looking for: Integrity literally means wholeness, and has an implication of honesty and morality. Someone who displays intellectual integrity would be someone whose intellectual works (writings, speeches) were free from dishonesty, shortcuts, plagiarism, etc. It would be someone who never uses faulty logic, and who is admired for making beneficial use of their intelligence. Opposites of intellectual integrity would be intellectual dishonesty, intellectual laziness, or being intellectually devious. I would re-write your example as: "The Nobel prizes and other major awards they have garnered over the years are a testament to their intellectual integrity."

Here in the expression "intellectual Integrity" the term integrity can refer to moralities and scruples as already said but I do think it could also refer to strength or prowess or sturdiness in terms of confidence and determination, depends on the further context though.

Intellectual Integrity = Recognition of the need to be true to one's own thinking; to be consistent in the intellectual standards one applies; to hold one's self to the same rigorous standards of evidence and proof to which one holds one's antagonists; to practice what one advocates for others; and to honestly admit discrepancies and inconsistencies in one's own thought and action.

this probably means something like: 有道德的智慧 (根據我的英語經驗，我直接翻譯成中文。。不知道這個說法順不順。。。） depends on if this means Nobel *peace* prizes or prizes for science. usually "integrity" is used to imply good morals. if this is for scientists who won the Nobel prize, I would find the use of the word 'integrity' confusing - since integrity is typically used to mean having good morals and being honest. integrity means united, but it very often implies "having good morals" sentence should be "Their intellectual integrity shows in the number of Nobel and other major prizes which they have garnered over the years. " - 'awarded' should not be used there, it is superfluous and makes the sentence less smooth.