It is possible that you can easily change to 'road' wheels, but also possible that it will not be a simple change.
It is also almost a certainty that changing the wheels alone will have virtually no effect, unless there is something wrong with your current wheels (damaged bearings, severely out of true). Changing the tires might make the bike faster - get something with a nice supple casing like a Panaracer Pasela 700X32C or so, and figure out a good pressure to keep them inflated at.

But as others have said, your position on the bike is very important. At speeds above ~15 mph or so, the vast majority of your effort goes into aerodynamic resistance, but also, a more stretched out rider position can often allow you to more effectively put power to the pedals. You can possibly flip your stem over so it gives you a few degrees of drop instead of a few degrees of rise, and/or take any spacers from underneath the stem and put them on top. If you have a 'riser' bar, switching to a straight bar will drop you down an inch or so. MTB style bar-ends can also give you a new hand position so you can bend at the waist a bit and get some more power into the pedals.

All this is assuming your bike already has an upright position but otherwise fits you well. If the bike is poorly set up for the rider, there are not really any minor tweaks you can do to make it better.

Lastly, you could try to find a drop-bar bike like an old ten speed or more modern road or touring bike. They are made for faster and more efficient riding on the road, so if you have one that fits you are probably going to be faster than you would be on a hybrid.