Yes, there is! And it's well worth doing!
Almost all the commercial river tours leave from the piers at Landungsbrucken, which is on the S-Bahn just south of the city centre. Also leaving from there are a number of local ferries that are part of the HVV network, which accept normal HVV tickets including day tickets:
The quieter option is to take the 72 or 73 boats, which leave from the east end of the Landungsbrucken piers, which take you a short way across the river. If you're in a hurry, these are the ones to go for
If you have more time, head to pier 1 and pick up the 62 ferry. On weekends, just look for the huge scrum, as there's no proper queues... Ferries are roughly every 15 minutes, so it's not the end of the world if you miss the first one. The 62 ferry takes about half an hour to make its way west along the river, giving stunning views (especially from the open air top deck) of the city, the city beaches, the port it. It runs to Finkenweder, which is a small area of the city just to the south west.
From Finkenweder, you can just stay on the 62 ferry, and ride it back to the city. This round trip takes about an hour. Or, if you have a little longer, get off and change onto the 64 ferry. This short trip takes you back to the north side of the river at Teufelsbruk. From there, you can either catch a bus to town, or enjoy a nice 15-20 minute walk through parks to reach the s-bahn at Klein Flottbek (the Botanical Garden).
The ferry map from HVV also gives a good idea of the routes and connections available.