Thomas Brinston (b. 1838 in Fermanah IRE) emigrated to Canada around 1850 with his parents, John and Jane. I don't know if there were any other family members that also made the trip. Not sure how to go about finding information about ships passenger lists.
There are no comprehensive lists of immigrants arriving in Canada prior to 1865. Until that year, shipping companies were not required by the government to keep their passenger manifests.
You might find an ancestor in correspondence of theSecretary of State on immigration for 1817-1857. There are
lists of immigrants, mainly Irish, and entry books re immigration, and assisted emigration to British North
America.
These must be ordered offline . Details are given at
http://olivetreegenealogy.com/articles/immcanpre1865.shtmlOn TheShipsList website they are including Quebec ship
arrivals extracted from contemporary newspapers.
See
http://www.theshipslist.com/ships/Arrivals/index.htmYou can also try inGeneas, the online searchable database of early immigration to Canada. Some records are free, others must be paid for, but the search and it's results are free. A search engine for InGeneas is at
http://olivetreegenealogy.com/ships/search_shipscanada.shtmlThere are also some passenger lists which were kept by shipping agents in the originating country. For example, the Passenger Books of J & J Cooke, Shipping Agents gives sailings from Londonderry to Philadelphia PA, Quebec, and St. John New Brunswick from 1847 to 1871.
For an index of clickable links to those JJ Cooke passenger lists online to date, see
http://olivetreegenealogy.com/ships/jjcooke.shtmlThe Hawke Papers, letterbooks of Chief Emigrant Agent Anthony B. Hawke are also available. They cover the years 1831 to 1892.
See the searchable database at
http://www.archives.gov.on.ca/english/db/hawke.htm