It doesn't specify that Nyquil cannot be taken. Since both medications can impair you, as long as you're taking them together at night... when you're going to sleep. I'd think you'd be okay. But you should ask your doctor regardless.
Levaquin can cause side effects that may impair your thinking or reactions. Be careful if you drive or do anything that requires you to be awake and alert.
-dizziness, fainting, fast or pounding heartbeat;
-sudden pain or swelling near your joints (especially in your arm or ankle);
-diarrhea that is watery or bloody;
-confusion, hallucinations, depression, unusual thoughts or behavior;
-pale or yellowed skin, dark colored urine, fever, weakness;
-urinating less than usual or not at all;
-easy bruising or bleeding;
-numbness, burning, pain, or tingly feeling in your hands or feet;
-fever, sore throat, and headache with a severe blistering, peeling, and red skin rash; or the first sign of any skin rash, no matter how mild.
-Less serious Levaquin side effects may include:
-nausea, diarrhea, constipation;
-feeling restless, anxious, or depressed;
-sleep problems (insomnia or nightmares);
-vaginal itching or discharge; or
-mild skin itching.
You should ask your doctor, but I have before...How much cough syrup is lethal?.