No. In short.
The longer answer: All apps in iOS are sandboxed, meaning they effectively have no permission to do anything other than that which they were written for, other than through specific exceptions through API calls to other parts of the OS etc as allowed. For example my Twitter app is allowed to access my contacts to use and set contact details as is deems fit. Many apps (in my case Instagram,Tweetbot and Youtube) are allowed to send tweets through the Twitter App, but in all these cases (or any case where one app access some data or functionality provided by a second app) this is not a system wide availability. Using said APIs an app can make portions of itself available to other apps, but unlike Quicksilver/Alfred/Launchbar on the Mac, this must explicitely be defined by the target app. So taking for example the "Due" app for iOS which is a reminders app, it has an API that will allow other apps to create reminders in it, but that is all that other apps are allowed to do, and in order to do so they must be aware of the Due API. As such the "Agenda" iOS calendar app has integrated the "Due" API, and is able to set a reminder in Due for an event created in Agenda. But it cannot search for existing reminders in Due, or just start Due (or close it) etc.
So in order for an app like QS to exist on IOS, it would need to incorportate the API of every other app that it might want to connect to, and be limited by the fact that a) not every app (in fact I'd say very very few) actually has an API, and b) even where they do, they are often tied down to very specific functionality.
A bit rambly, but I hope this answers your question.
Just got an upvote on this, and re-read, and there is more the the answer than when I first made it. There are a few apps these days that are trying to get closer to what the OP wanted, by effectively collecting and allowing the use of as many of these 3rd party APIs as possible in one place. One such ass is Launch Center which collates APIs for nearly 50 apps, allowing you to do all sorts of actions from a single place - it's no quicksilver replacement, but it is an effective way to get at particular app functions directly rather than having to open the app and navigate to the function manually. - 1 example, is using clipboard contents for all sorts of things - tweet using clipboard contents, new task from clipboard contents etc etc. Very useful.