The only Alexander Quay (for your time period)(Quee, Kwee, and in the old records McQuee "indicating Scotch origins") that I have found is my 5 x's great grandfather Alexander Quee of PA who was b. ca 1721 and served in the French and Indian War here in NY State for two time periods in the 1760's. He married a Margarita and they had John Andrew Quay, bpt. 1767 in Albany, NY who settled in the Helderbergs and m. Maria Zimmer in 1797. Alexander's father was also an Alexander who was in PA in 1709 (found an old will in which he is mentioned). Alexander m. twice. According to my mother, a Quay, a couple of stories circulate on Alexander Sr.'s birth place. Mom stated that a couple of brothers came from Edinburgh, Scotland, another family member said they were French Huguenots. There probably is a little truth in each as the early Quays did marry in with other French Huguenots - Du Trieu's (Truax) and La Grange, etc.
A lady I corresponded with in England declares they were the McQuays - Irish, another said they all came from the Isle of Man. Another says that they were certainly Scotch and I'm inclined to think that perhaps they were Celtic, went to Scotland, stayed a couple hundred years, many went back to Ireland and then to America. I have tracked a Qua family from Ireland, they all have similar stories.