It would be perfectly possible and not too difficult. Every post containing
[quote author=<ignored_used>]
could be hidden/ignored.
The only issue I see is that a post could be from someone you care about and could include useful information. To solve that only the quote itself could be removed/hidden while showing everything else.
Edit: Actually it would be easy to create a userscript that does that. I'll try getting some time to work on that soon, unless someone writes it first.