He is listed on the 1830 Census with 3 boys and 1 girl and the 1840 census with 1 boy.  He married 8 Jan 1818 Pulaski Co., Betsy Whiteaker.  (Pulaski Co. Marriages, Book 1, pg. 16)

Also in  Marriages Book 1, pg. 16:
23 Sep 1839  Brinson, Giles W.  Adkins, Nancy   Rial Adkins
con. Zebulon Brinson & John Brinson   Jas. Adkins, b.f.
wit. Benj. Rial & Jas. Adkins

5 Apr. 1841 Brinson, Jas. R.   Adkins, Jemima   Rial Adkins
con. Zebulon Brinson, g.f. & Jas. Adkins, b.f.
This is the only proven child I have found.
Believe Giles W. above is also son.

Believe Zebulon is the son of John Brinson & Belinda Taylor.  Married 12 Dec 1795.
No Proof.

Deed Book #14, pg. 341 dated Oct 1850 lists heirs of John Brinson,
deceased:
John H. Decker & Dicy 
James Hargis & Polly
James Hargis, Jr. & Catherine
Isiah Bray & Phebe
Amos Chaney & Temperance
Singleton Earp & Hannah
Lorenzo D. Phelps & Sarah
Taylor Brinson & Nelly

Note Zebulon is not named.  I have marriage records and some census records of those listed above.

What is the connection of 
Jonathan Brinson b. ca 1820 Pulaski Co. d. 4 Jan 1898 PC 
m/1 Jane Phelps 15 Mar. 1841 
m/2 Martha Jane Meece 24 Oct 1872? 
Some say he is Zebulon's son.  Need Proof.

I have Pulaski Co. Marriage Books 1-4, 1810-1880 Censuses, 1799-1800
Taxpayers.  I don't mind looking up small amounts of info.