Düsseldorf lies in the mostly flat industrial corridor of the lower Rhine. There might be some hiking trails outside the city, but don't expect any grand vistas.
The closest you'll find to anything remotely like the Swiss Alps in that region is either the Sauerland east of Köln, the Westerwald to Köln's south, or the Eifel (including the National Park) south of Aachen. But think rolling hills and low mountains instead of towering peaks. I've never hiked in these regions, so I don't have any specifics.
When you arrive in Düsseldorf, go to a bookstore and look for hiking guidebooks. If you won't have a car, the Rother Wandelführer guides are particularly useful, because many of the recommended hikes in these books begin from a train station.