There are 2 ways to access web content -
QtNetwork module and
QtWebKit module.
QtNetwork allow you to "low-level" control of HTTP, but in this case you need to "manually" analyze received documents and, if needed, download their parts (images, flash objects, included css and js files, etc).
QtWebKit provide more easy access - you just use some methods (load|setUrl) and it download all content for you. You can specify some options - allow/disallow auto downloading images, enabling/disabling java/javascript (yes, it's one of the major nice things - QtWebKit can process js), etc. After it you can use methods toHtml|toPlainText to get source HTML or rendered text.
But in both modules there are some problems with access to HTML document's elements - WebKit have HtmlDocument and API to manipulate it, but in QtWebKit there are no such API (I hope - yet). (I still can't find Qt's native methods for working with HTML, have to use regexps).
I think, QtWebKit - is what you need, but it depended on known only for you factors.
NB: If you'll use QtWebKit, even if your crawler is console application, you have to link with QtGui module - QtWebKit depended on it.