There are different reasons for birds to bite / nip:
= A bonded bird nipping lightly as part of preening behavior. This is normally light nipping that travels along a line or area of the skin. If it is painful to the human, a soft high-pitched squeak without any startling movements can tell inform the bird that its painful and they will be more gentle naturally; as if grooming a flock mate in the wild.
= A bonded bird biting or nipping harder because it didn’t get what it wants / you did something ‘wrong’ / it was not allowed to bite something [ such as a computer wire ] because you are watching out for its safety. Here the bird is showing its displeasure ON PURPOSE so a firm ‘NO’ without too much volume or movement acts as a warning. Ease the bird away from you or leave the room if it continues biting, or remove the object that it wants. To be firm without being violent or aggressive to the bird.
= A bird unbonded to a certain human biting out of fear. This can come as quick sharp bites or even hard chewing with the full intention to break skin / cause pain to the perceived threat. In this case, reacting loudly and violently, even destabilizing his perch on an arm, and then leaving the bird alone is the wrong move. Not only does it reinforce to the bird that the bitten human is correctly perceived as a threat / predator / unreliable perch, but that biting makes the threat go away eventually. This reinforces wrong behavior and needs both patience and pain resilience to correct.
It sounds like he is biting her out of fear. Perhaps you need to figure out what causes him to bite [ a certain cue, body language, color? ] or try to cultivate the relationship he might have with her. If she needs to pick him up, using a suitable sized stick or a perch, and asking him to step up would be safer. Another note is to ensure she does not to tease him if he's in the cage [ie. putting her fingers near the bars and testing if he reacts] as many birds are territorial of their cage. He might view it as a predator attempting to invade a safe space. Offering treats through the cage bars may help make her seem more friendly. Talking to him soothingly as if talking to a toddler.
Would be good to have some play sessions with him with your guidance to learn how to handle him and to build trust. Have her feed him fruits or treats, either on a skewer stick, or later by hand when he is more comfortable. Try watching some youtube videos that explain parrot behavior, prioritize those that are specifically parrot training channels and not random youtube celebrities showing off their birds .