Will nyquil help get sleepy even if I'm not sick?

Yes, but why? Why take a pain reliever, cough suppressant and antihistamine just to get sleepy. There are much better sleep aids than nyquil.
NyQuill. Yes it will as it has alcohol in it. It is not a good idea to use these medications to get to sleep. They can very easily be misused. Get into a proper sleep hygeine plan - eat early, no tv in bedroom, no coffee or alcohol at night, quiet dark room, comfortable bed etc... These usually work, if they do not then see your doctor before you start using your own medicinals.