Sounds like it may be Pea beans. They are a bush dry bean, with small pods and small round white seeds. Back around mid-century they were pretty much a standard variety in some parts of Canada, northern Minnesota, and other areas.

Navy and Great Northern beans surplanted them, primarily because the beans are larger.

Fearing Burr talks about them in his 1863 "Field & Garden Vegetables of America. Among other things he says, "The ripe seeds of the pure variety are quite small, rounish-ovoid...and of a pure, yet not glossy, white color.

"As a garden variety, it is of little value, thought he yound pods are crisp and tender. It is cultivated almost exclusively as a field-bean."

Pea beans are said to have originated with the Iroquoian tribes of New York.

How small were the beans? Very tiny? If so, they may have been rice beans.

Jim

Here is a link that might be useful: Rice Beans

Here is another rice bean link. Rice beans come in many colors, usually not white, but these are. I have never seen anything saying they were used as a baked bean.

There has been some changes in the taxonomy. They are now put in the genus Vigna, not Phaseolus.

Jim

Here is a link that might be useful: More Rice Beans

I'd think that anything Vigna would require more heat units than Quebec has to offer. That means that the pea bean is much more likely to be what you are looking for. NOW you just need to find seed for it. Here's a bit of advice: if you go talking to oldtimers in your search for seed, don't turn down anything else they offer you. You'd be amazed what you can turn up. (You might want to talk to someone on an Indian reservation.) Good luck! Donna

The bean may have been a Flageolet.

I think that the variety you are looking for is called Haricot nain Comtesse de Chambord. Quite a rare bean but the link shows that it is still grown in Quebec.
Rose-Marie

Here is a link that might be useful: Haricot Comtesse de Chambord

Now that was interesting for me. I got my beans from my friend in Canada. The "pure" beans are a deep chestnut brown with a small white eye. In the "mixed" are some just like in that picture! They are that white with a Redbrown "man figure" on them. Those really caught my eye when I opened the package. I had planned on planting all of those to see if they produced beans like themselves as the figure is so striking.
dorothy in wyoming

The bean with the 'man figure' is known as 'soldier bean' or 'European soldier bean' because it is thought to resemble a soldier in red uniform. It is a large bean.

Jim