Conway County. Conway County was formed on October 20, 1825, from territory taken from Pulaski County and was named for Henry Wharton Conway, a member of the Arkansas territorial delegation to Congress.Morrilton is the county seat. Conway County landscape is rolling farmlands, forested ridges, isolated mountains and lakes.onway County. Conway County was formed on October 20, 1825, from territory taken from Pulaski County and was named for Henry Wharton Conway, a member of the Arkansas territorial delegation to Congress.