If a deputy lies under oath during his testimony, is it perjury? What would make it perjury?

Yes, it could be considered perjury to lie under oath. This is true for members of law enforcement just as it is for anyone else. Proving that perjury has occurred is another matter and has a couple of requirements.
First of all, it has to be proved that the lie was made intentionally. Simply getting information wrong or being inconsistent and testifying to that is not perjury. There needs to be clear evidence that the person giving the statement was aware that he or she was making a false statement. Proving intent is always a tricky thing. You would need some sort of evidence showing that they intended to lie while under oath. This is why there are so few actual perjury convictions.
Also, keep in mind that a lie is not necessarily perjury if it is immaterial to the matter at hand. It needs to have an influence on the outcome of the case or deposition. For instance, if a deputy states that he or she had quit smoking, but this was untrue but also completely irrelevant to the testimony, then it would not be considered perjury. The statement must directly impact the testimony and its outcome to be considered perjury.
https://definitions.uslegal.com/p/perjury/