Yes, it does occur with verbs, but this is very colloquial and I would also say it was non-standard*; also, the meaning seems to be different. In your examples, it emphasizes the verbs, if anything. It doesn't mean the same as the standard modifier 'fairly' which means "rather; to a moderately high degree". For that meaning you need 'fairly much':
We've fairly much finished. [Perhaps not 100%, but we've finished the important parts]
'Quite' of course has two meanings, "rather, fairly" and "completely". In your examples, 'quite' is not really idiomatic in the first, and could be used in the second, with the meaning "completely". In effect, this has the same meaning as the colloquial 'fairly' here. It's not "almost, mostly, to a moderate degree", but rather it emphasizes the destruction. Another example would be:
It (the news) quite/fairly astonished me.
It might also be restricted in verbs: snapping, destroying, and astonishing are absolute; you can't really do them to a moderate degree. (If you snap or astonish moderately, the strong words 'snap' and 'astonish' are the wrong choice of word for it.)
* But the example below, 'the hall fairly rang with applause', sound more standard to me, so I'm not sure.