Have you obtained a copy of William F. Brown's pension application file? Did he ever live in Claiborne Parish? I found the following listing in the Confederate Pension Application Index:

BROWN, WILLIAM F.
Parish: CLAIBORNE
State: LA
Branch: INFANTRY
Unit: 28TH REGIMENT (GRAY'S)
Company: CO. D

If the above listing is your relative and the pension was granted, the file may contain information regarding his death.

I also found the following listing in the Louisiana Statewide Death Index 1900-1929:

Name: W. F. Brown
Death Date: 26 Jan 1922
Estimated Birth Year: 1833
Age: 89 years
Parish: Claiborne
Certificate Number: 2421
Volume: 5

The above W. F. Brown could possibly be your relative. You would have to order a copy of this death certificate to see if more information is given. The death certificate will usually name the spouse and/or parents of the deceased, but not always. The place of burial is usually given, also, but not always. Did he ever live in Claiborne Parish?