I think your grandfather might have been born in Oklahoma.
Take a look at these records from FamilySearch. In particular, there is the marriage record between a Robert Thomas Tunin and Abbie Alice Brooks. It gives Alice's parents names as well.
It looks like someone has created a relevant Tunin family tree on Ancestry.com for your family. Here is a quote from a story posted there:
In 1893, Willard Barton and Lillian May (Barney) Tunin, came from near
Cherryvale, Kansas, to homestead in Dewey County, Oklahoma; in a
covered wagon with two small children: Viola May, 3, and Mary Almeda,
a baby. Five more children were born on the homestead, making a total
of 8 children, one of whom died in infancy and was buried in Kansas.
Those born in Oklahoma were: Rosa Elvira, Georgia, Gertie, Robert
Thomas and Annalina. Almeda died in 1900 at age 6; Georgia in 1912 at
14 and Viola May in 1950 at 59 years.
While I haven't checked how well sourced this information is (I did not find a 1910 or 1920 census for them), it seems sufficiently detailed to have a bunch of clues to pursue.