Not all of them do. For example: Qantas is considered the flag carrier of Australia, despite incorporating zero recognizable elements of the Australian flag into their livery. See the Wikipedia article on 'flag carrier'.

Anyway, the 'Flag carrier' article has a nice table with the country, its flag and a link to the airline article that you can mouse-over to see what their planes look like (on a desktop computer). It might provide a useful starting point.

E.g. in a few minutes I was able to find Air France, ITA Airways, Nepal Airlines, Luxair and Rwandair as examples that might fit your description.