Peter Gaydos is listed as a widower in the 1930 census with the 4 surviving children, Elizabeth, Joseph, Julia and Frank. All the children list West Virginia as the place of birth. Justina died Dec 20 1929 and is buried next to Peter in Louisiana. Both tombstones have the Slovak spelling Gajdos - see findagrave.com. Also, in the 1930 census, there is a John and Nellie Gaydos living next door to them - do you know of any relatives who also moved to Louisiana?
I found a Pete Gados in the 1920 census. His wife is listed as Kate, but I think it is your great grandparents as the children are Lizzie, Katos, Joe and Julia. They also have a boarder living with them - Paul Dudas. Given that his last name is Justine's maiden name and the children's name, this is most likely them. They were living in Logan County, West Virginia. A search of West Virginia vital records shows that a Peter Gajdos and Jessie Dudas Gajdos of Logan County lost a daughter, Mary, in 1923 due to premature birth, which helps to confirm that this 1920 census record is indeed them.
What is confusing is in the 1930 census, Peter says he came to America in 1907, in the 1920 census he says he came in 1910 so I have not been able to get immigration info.
You might try to get obituaries from Livingston Parish Louisiana for Peter and Justina to see if you can get any further info. Maybe an obit would list a town of birth.
Good luck with your search.