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One of the things that's always been a minor peeve to me on LiveJournal (and then, later, on Dreamwidth) is the Forrest Gump box-of-chocolates nature of the Quote button. (You never know quite what markup you're going to get.) For LiveJournal, I long ago found a Greasemonkey script called LiveJournal Blockquote, which made the Quote button consistently generate a <blockquote> tag. But it doesn't work on Dreamwidth (and appears to have vanished from userscripts.org anyway).
That's why I wrote an improved version today, called LJ Quote Fix. It works on both LiveJournal and Dreamwidth, should work on any site based on the LiveJournal codebase, and has provisions for customizing and styling your quote markup using CSS (though at this time there isn't a UI for it, other than Greasemonkey's 'Edit' button.)
By default, it formats quotes like this. (Well, mostly like this, anyway. If you're reading this in my style, the light-blue background comes from my journal's custom CSS.)