Hi Norman,
There are quite a few inns/pubs currently listed in Liphook (Deers Hut, Royal Anchor, Caladonian South, Green Dragon, Links, Rising Sun, William Wynne)however, finding which, if any, existed in the 1700's is proving difficult. The Deers Hut dates from the 1600's but was originally a hunting lodge.
There is a Bramshott and Liphook Preservation Society that can be contacted with enquiries at 12 London Road, Liphook, Hampshire GU30 7AN (can't find a website for them).
Is Nicholas Elkins the particular Innkeeper you're interested in? If so, is this website of any interest:
www.lewcock.net/genealogy/ It contains Nicholas and descendants.