It is quite possible your side of the McQuay clan came from Scotland.
They may also have come from Ireland a great number of people with descendants in America seem to come from there!
There were a great number of McQuay's in Galloway Scotland from 1600 to 1850 probably a good place to start if you were keen on looking.
From myself looking on the internet there seems to be three main groups of the Mac clan in Scotland in the period 1600 to 1850.
Finding things about your family in the UK is not easy all the records are not on line! But were working on it as they say.
It was not unusual to find a corruption of the McQuay name for example McQhae different spelling same pronunciation.This is a common name in Galloway between 1600 to 1850. You have to remember literacy was not a common thing in those days.
To make any progress in tracing your ancestors in UK you must know really when the member of your family emigrated, to have something to go on at the public records office or parish records.
I wish you the best of luck sorryI cannot help more without more specific info.
The literal translation from the Gaelicof McQuay is "son of mac" so if you see a name of Quay it is not likely to be directly linked unless somebody deliberatley changed their name.