I recieved the following response from Vicks, the company that maufactures or offers NyQuil:
Thanks for your question regarding what the "Quil" stands for in NyQuil and DayQuil. As is sometimes the case with acquired brands, that specific information does not exist in our Archives. P&G acquired Richardson-Vicks (RVI), the company that introduced NyQuil, in 1985. However only a fragment of the historical files from RVI were ever turned over to P&G. In a 1999 magazine article at the time of the brand's 30th anniversary, the author conjectured that NyQuil was based on the term "nighttime tranquility." Although I can't confirm it, it would seem to be a logical guess.
My apologies that I can't provide a definitive answer.
I thought that this would be the response, thank you all!