The brief bio of William at www.manxnotebook.com/people/law/whqllm.htm states his family were in the watch trade - at the address quoted in 1881 is Robert Quilliam Wholesale Watch Manufacturer Emp 60 Men age 49 b. Liverpool married to Harriet Burrows - looking for Robert in IGI gives no leads as to father - there are several Quilliam families in Liverpool certainly from the 1790's onwards looking at the IGI
(the 1881 census gives W H age as 24 which agrees with the 1856 birth date - however no match in freebmd I can find. A 25 year gap between children is possible but usually indicates a 2nd marriage or an early age at marriage for the woman