Typically in people's lives, conflict between generations shows up in the two generations not understanding each other's values and mind set. I think you can see this very clearly in this story (although it is not just a generational thing).
In the story, Mrs. Johnson represents an older generation. She is of the generation that grew up during the Depression, when there really was very little opportunity for blacks in the South. So she values things like quilts that are practical. She does not understand the new generation's ideas about old stuff being quaint and important as folk art.
By contrast, Dee (Wangero) has had a much more privileged life as a member of the younger generation. She does not understand how hard her mother had it and therefore doesn't really understand what things like the quilts mean to her mom.
Alice Walker was fascinated with women of her mother's generation. She expresses this in "Everyday Use" (Mrs. Johnson), The Color Purple (Celie, Sophia), and her poem "Women":
They were women then
My mama's generation
Husky of voice- Stout of Step
With fists as well as hands
How they battered down doors
Starched white shirts
How they led Armies
Across mined fields
To discover books desks
A place for us
How they knew what we must know
Without knowing a page of it themselves.
These women, Mrs. Johnson, and the women in The Color Purple (Celie, Sophia) were uneducated, even illiterate because they lived in the Jim Crow South. Their primary goals were to educate their daughters' Civl Rights generation: Dee, Alice Walker, Adam, Olivia (in The Color Purple). Their role as matriarchs was find educational resources for their daughters that they never had.
The problem is that on Civil Rights era daughter, Deem takes her mother's hard work and self-suffeciency for granted. She exploits her hard work and struggle as an emblem of African nationalism instead of sacrifice.