Latin translation for "integrity"

That's partly because there are various meanings for "integrity" in English.

"Veritas" is Latin for "truth".

If by "integrity" you mean "adherence to moral and ethical principles" or
"soundness of moral character", then I think the word I would use is "honestas".

Note that the word "integrity" is derived from the Latin word "integritas", which
means "completeness" or "soundness" and by extension could also be used to
mean "integrity" as we have defined it above.