The only Web-based solution currently offered by Quicken is Mint. You can learn more about Mint at the Quicken Web site.
They have also just released a Quicken Mobile App usable on iPhones and Android devices. It links to the 2013 version of Quicken for Windows. I don't know whether this would count as an Android device, but even if it does, you'd still have to also be running Quicken on a separate Windows PC.