The incident happened early on Tuesday at Dahagram border area of the Upazila, BGB's Lalmonirhat 15th Battalion Commander Lt Col Ahmed Bazlur Rahman Hayati said.
He identified the dead as Hazrat Ali, 27, of Dahagram Union's Fakirparha.
Hayati told bdnews24.com that BSF personnel had opened fire when a group of at least nine Bangladeshis crossed the border and entered Indian territory illegally to bring the cattle.
"Hazrat Ali died on the spot. We have sent a letter (to BSF) asking them to hand over of his body," the BGB official said, adding that he was on his way to the area where the shooting occurred.
Dahagram Union Council Chairman Kamal Hossain said the rest of the group had managed to flee while Ali got killed. BSF personnel took away his body.