I have a Quine family at the other end of Strand Street (No 93 in 1771) - Edward Quine (b: 1813), my 2 x Gt Grandfather, who was also a boot and shoe maker in 1871. He married. Esther Jane Oswald in 1849 (a daughter of Robert Henry Oswald, ex Surgeon to Household of Manx Leut. Governor) and they lived originally in Back Strand Street.
He must have had a good business as he managed to retire before 1881. Nobody else became boot or shoe makers, though, and I can find no relationship with your Quines as yet. Does anyone know of any surviving apprenticeship records for the Island?
Roger