Is this just the dates of the name change or the dates of the "County" changes in Quebec? Prior to 1763 New France used a system of land grants/Seigneuries in 3 administrative areas (Quebec City, Montreal, Trois Rivieres). After the British occupation in 1763 they introduced the county system, and many of the larger Seigneurie names were used.
A good reference for the Seigneuries is here:
There seem to be a few missing at the western edge:Seigneurie de L'Île-Perrot, Seigneurie de Vaudreuil, Seigneurie de Soulanges and Seigneurie de Rigaud
The seigneurial system (which after 1773 included some English land grants) was finally abolished in 1854.
In 1774 the name was officially changed to Québec. Note that at this point there was still land in the US that was part of Quebec.
in 1791 Québe cwas divided into Upper and Lower Canada
in 1818 the Canada/US border defined as the 49th parallel
in 1841 Lower Canada renamed Canada East. Canada East and Canada West referred to as "Province of Canada"
in 1867 renames Province of Québec
I need to claim ignorance on the official usage of British North America. I believe that it is the encompassing term used from 1780's to 1867 to represent the NA territorial lands (including parts of the present day US).
Wikipedia has an article on BNA
in 1982 Quebec moved from Counties to regional county municipality (RCM) or municipalité régionale de comté (MRC) and equivalent territories (TE) which redefined a number of boundaries. A list of historic counties can be found here:
Note that their boundaries also changed occasionally over the years.
There is also Rupert's Land that existed from 1700 to 1870ish, eventually combined with the North-West Territories in 1870. The NWT encompassed parts of Ontario and Manitoba through 1905ish.
Personally, when referring to anything that is located in Quebec, I just use the Quebec, Canada irrespective of date rather than the "Lower Canada"/"Canada East" designations.
I hope this helps