The McQuartag family was nowhere to be seen in the Treen of Dromrewagh in the c1490 rent roll, but John was definitely there in 1513.
The derivation of Corjeag from MacQuartag would seem to be in line with other families dropping (or having dropped) the Mac in the mid-sixteenth c., From the early 17th c. we have the succession of tenants of Balla(g)lonna, some leaving wills. The idea that certain Manx families had English Gentry origins was one that AWM seems to have been fond of.

He wilfully/negligently ignored the baptismal date of his own direct ancestor Robert Moore in the family bible in his own possession in order to claim descent from a different (more important??) branch of (ironically the same) family. He latched onto an English family of the same name (Moore of Bank Hall, Lancs) and this was evidently what he wished upon the Corjeags.

There have been no Corjeags left on the Island since my grandfather's 1st cousin John Haddon Corjeag died, but of course there are still the Cavendish descendants of Patrick C.

I am also descended from the Corjeags of Ballakilley, Kirk MIchael. They died out even longer ago, but my relative who still lives at Ballonna was at one time in contact with a family called CORGEE in the USA who were definite descendants of them. Over there since the 18th c.

Our (Balonna) branch occasionally lapsed into using Cavendish, as will be seen from my 3rd gt gr father William's marriage in Patrick register.

Good luck with any further discoveries.