For a start, there are over 100 Quilliams in the 1891 census (and there is at least one variant), so the name was popular on the Island. There are about 40 Quilliams in the 2000 telephone directory but that is always an under-estimate as 40% of the population is ex-directory or on a mobile.
You will have to ask your father for the details of his research and then write to the Registry for the certificates, so that you can build a picture of your family. If the gentleman concerned was "direct" family, you should find his name quite quickly or, at least, identify several candidates. Then you will have to use the telephone directory or the electoral register.
You might have to pay a researcher to do the ground-work on the Island for you, or wait until you have a few days to spend on the Island - lovely at any time!
I hope to be on the Island next week, but I will be very busy and if I don't find an internet cafe , I will not be reading my account! However, if you have an easy query, I might be able to oblige.