Well to be honest I don't know the details of what any of those functions do, just thought since (I believe) it's doing something like what you're trying, maybe something would be useful.
As it turns out, it looks like "strip_quotes" (which is in functions.php) strips [quote] tags out of the text. So I guess it had nothing to do with your quote problem, sorry about that.
Maybe you can use the php str_replace function to replace ' with \' (put a backslash in front of it)? I'm afraid I'm not much help at this point - maybe worse than none if I lead you astray.