The word "good" is an adjective and therefore modifies nous (in this sentence, the noun "it").
The word "well" is an adverb and therefore modifies verbs, adjectives and adverbs. In the sentence "It's (going/doing) very well" the word "very" acts as an adverb because it is emphasizing how strong the word "well" is expressed.
But when describing health and well-being, we use 'well' as an adjective. If someone asks 'How are you?', in the UK we would traditionally say 'I'm very well, thanks', and the tendency to say 'I'm good, thanks' has come from the US in the last decade or two, or so I've been told. So in this case, maybe the person is responding to the question of 'How is your dog?' 'It's very well.'