Do you have access to the 1910 U.S. Census?I found a "Robert and Ella Elrod" living in Bark Camp, Hall County, Georgia.Robert listed his age as 32, which would make his year of birth about 1878. They listed 3 sons:Lee, 9; Joe, 7; and Pinkney, 5.Joe would have been born about 1903.This could be the Joseph M. Elrod you are searching for.
In the 1880 U.S. Census for Bark Camp, Hall County, Georgia, there is a "Joseph M. and Nancy Elrod" family listed with 4 children: Elias (son) 8; Frances E. (daughter) 5; Robert (son) 3; and Mattie E. (daughter) 9 months.This Robert would have been born about 1877.It is likely the same Robert who is listed in the 1910 Census.
The Joseph M. Elrod you are searching for could have been named after his grandfather.
Hope this helps.I am an Elrod descendant, but this is not a family I am familiar with.