I don't think giving your dog Nyquil is a good idea but you could try a very small dose of Children's Benadryl. My vet has told me to give my small terrier benadryl for allergies if needed and my brother has given his dog some on long road trips to keep him calm.
Why would you even think of giving a dog Nyquil? Nyquil has a very high alcohol content and is definitely NOT good for dogs. Do NOT give it to him. What are his symptoms that you think he needs it?
Ghilly