hi myself1, you have done much for New Brunswick roots I am trying to help
Does a passenger list exist for 1818 it might show a group from one location in Ireland left together
I tried 
http://www.dippam.ac.uk/ied/16-01-1818   Ship Hugh Wallace, New Brunswick to Portaferry.   Central Library, Belfast   Newspapers (Shipping News)
30-01-1818   Brig Hugh Wallace, Belfast to Miramichi, Nova Scotia.   Central Library, Belfast   Newspapers (Shipping Advertisements)
20-02-1818   Arrivals Departures & News of Various Ships   Central Library, Belfast   Newspapers (Shipping News)
25-05-1818   Brig Fame, Belfast to St. Andrew's, New Brunswick.   Central Library, Belfast   Newspapers (Shipping Advertisements)
14-06-1818   Brig Ganges, Belfast to British America.   Central Library, Belfast   Newspapers (Shipping Advertisements)
15-06-1818   Ship Endeavour, Belfast to St. Andrew's, New Brunswick.   Central Library, Belfast   Newspapers (Shipping Advertisements)
27-07-1818   Brig Lion, Belfast To St. Andrews, New Brunswick.   Central Library, Belfast   Newspapers (Shipping Advertisements)
28-07-1818   Brig Ganges, Belfast to St. John's, New Brunswick.   Central Library, Belfast   Newspapers (Shipping Advertisements)
01-12-1818   Ship Regent, Liverpool to Canada.   Central Library, Belfast   Newspapers (Shipping Advertisements)
29-01-1819   Letter From William Mitchell, Master Of The "Mary Ann".   Central Library, Belfast   Newspapers (Extracts)
05-02-1819   Snow Dorcas Savage et al, Portaferry to New Brunswick.   Central Library, Belfast   
comment these are notices of ships leaving but not passenger lists
Your Gibson line
Alexander Gibson b 1775 Lanarkshire, Scotland
Married: 1794 Newtown-Ardes,Down,Ireland
Spouse: Janet/Jennette/Jean Moore b 1775 Scotland
children
Stewart Gibson born ? In either Scotland, Ireland or posibely New Brunswick, married Rebecca Jane Evans. Stewart and Rebecca lived in New Brunswick Canada,their daughter
children
Margaret King Gibson was born 1837 in ST.James Charlotte County New Brunswick Canada,she married John Russell Blair in 1863.
staying on that Newtownards linksee 
http://www.newtownardshistory.co.uk/family-gibson.htmbirths
Gibson, John - 1795 - (Father Alexander Gibson 1775) - (Mother Janet Moore 1775)
Gibson, James - 1800 - (Father Alexander Gibson 1775) - (Mother Janet Moore 1775)
Gibson, Margaret - 1804 - (Father Alexander Gibson 1775) - (Mother Janet Moore 1775)
Gibson, Stuart - 1812 - (Father Alexander Gibson 1775) - (Mother Janet Moore 1775)
A recent {2013} e.mail from Heidi Russell informed me that Alexander Gibson her GGGG Grandfather was born in Glasgow, Scotland on 20th. August 1775
marriages
Gibson, Margaret - 1786 - Married John Orr
flag-scotland.   Gibson, Alexander of Scotland - c1794 aged 24 - Married Janet Moore
Gibson, Robert - 26th. April 1809 - Married Rachel Patterson - {Ref. = LDS72222120/150} - Also the Belfast Monthly Magazine
the evans entries are all much more recent
comment1. are you the  H......R...... above?
2.  the estimated  marriage date for a Alexander Gibson m a  Janet Moore is 1794+24 = 1818 location unknown yet its on a newtownards board. 
Church  records for Newtownards the surviving church records are listed 
http://www.rosdavies.com/CHURCHES/K_W.htm#Newtownards3. You interchange Stewart to Stuart b 1812[-1814] , i understand that vision of records of those days 
4.  the website compiler Derek Beattie died suddenly in 2016, his 10 special  friends have resurrected  his website however his sources for each entry  aren’t obvious to me yet.
I stop here for encouragement or advice