The short answer is I doubt it. The long answer is I doubt it, but even if it is, it's probably a bad idea.
Quoted text is contained within the post, which is why you see it, it's also why quotes such as the one below can be made despite not being actual posts. The ignore function doesn't delete anything, it just hides posts. Reading and removing/hiding sections of posts goes way beyond that.
I like big butts and I cannot lie
The quote will link back to a post that wasn't made by Sir Mixalot and does not contain the quoted text.
It may be possible to install a script that hides posts that quote posts that would be hidden (I.e. By ignored users). I genuinely do not know. But it would probably slow the forum down, considering what it would have to do (read every post on that page, check if it contains any quotes, then check if any of those posts refer to posters you have on ignore and the hide those posts). It also means that posts containing quotes of more than one user (e.g. You and someone on your ignore list) would be hidden, even though you don't have me on ignore and you may well want to know what I said in response to you. It could also mean that posts quoting a post that quotes a post by a user you have ignored would be hidden (because it's quoting a hidden post) and so forth.
An alternative could be removing the text contained within a [quote] block if it refers to an ignored user. But that leaves you with a context free reply to the quoted text and fails completely if not all of the quoted sections of a hidden post contain the post ID. For example if I broke the post down into multiple quote blocks to respond to parts of the post individually but didn't paste in the ID into any quote blocks I made you'd be seeing parts of a post, but not all of it which would be annoying.
Cue someone from VScope telling me that it's actually very simple and can be done straight away with none of the problems I foresaw.