Ahmad Ghamloush was unintentionally killed and another man wounded when soldiers fired warning shots into the air to try and stop fighting between followers of the rival movements.
Speaking to Reuters, the official said the incident took place in the southern town of Jabaa, near Nabatiyah.
The dispute stemmed from an argument over the location of banners and signs.
"When they resisted the patrol, some of the patrol fired warnings into the air. Due to this fire, two were shot. One was killed and the other injured," the official said.
"It (the death) was unintentional. The aim of the shooting was to keep them away from each other," he added, saying dozens of people had joined the clashes.
Hizb Allah officials had no immediate comment.
According to the army official, it is the second incident in under a month where an argument between Hizb Allah and Amal followers has resulted in death.
Earlier this month, one person was killed in clashes between the two groups over the placement of signs in Beirut's southern suburbs. Authorities made three arrests in that case.