How about the below entries, found on Ancestry.com?

(1) You wrote for 1910 entry "Hezekiah Carter b. ~1874 m. Hannah J. b. ~1885 (2nd marriage for both, this marriage for 5 years)"

That puts their marriage about 1905. Though the age may differ from the 1910 age noted, on the below entry note Hannah is listed as "Mrs. Hannah J. Fisher". NC marriage licenses ask the parents' names. If the below is the correct couple, you should be able to acquire a copy of the license and see if parental information was completed.

North Carolina Marriage Collection, 1741-2004
about Hezekiah Carter
Name: Hezekiah Carter
Gender: Male
Birth Date: abt 1875
Age: 30
Race: Colored (Black)

Spouse: Mrs. Hannah Fisher
Spouse Gender: Female
Spouse Age: 22
Spouse Race: Colored (Black)

Marriage Date: 30 Dec 1905
Marriage Location: Beaufort
Marriage County: Carteret
Marriage State: North Carolina

Source Vendor: North Carolina State Archives

(2) Possibly a death entry for your Hezekiah.....

North Carolina Death Collection, 1908-1996
about Hezekiah Carter
Name: Hezekiah Carter
Death Date: 26 Jun 1958
Death County: Craven
Death State: North Carolina
Death Age: 78

Birth Date: 1880

Race: Black

Source: NC State Archives. North Carolina Deaths, 1908-67

If this you think he is yours, you can order an uncertified copy from Craven county register of deeds for a quarter. Perhaps a parent or birth location is recorded on it that could be used for further research.
Here's the Craven County Register of Deed's website, which also allows you to search some of their indices online - http://www.co.craven.nc.us/departments/reg.cfm.

Happy Hunting,
Damita