RNZ understands the video was posted to the boy's Instagram story. RNZ has opted not to publish the full footage.
MPI said it had received complaints.
"The behaviour in the video is unacceptable, and we will be making enquires."
Police said they had also received an online report on January 25 regarding the video posted on Instagram, which appeared to show an octopus being beaten.
SAFE chief executive Debra Ashton told RNZ she thought it was a "senseless, disturbing and violent attack on an innocent sea creature".
"SAFE has laid formal complaint with the Ministry for Primary Industries. They have told me that this has been passed onto their investigations team.
"We hope that members of the public will come forward to help identify these people and that they will be held to account for their actions.
"These people need to be stopped before they carry out these types of attacks on others."
The Department of Conservation declined to comment saying it was a matter for MPI.
A spokesperson for the Tauranga Sport Fishing Club said they were aware of the video but did not believe the pair were affiliated with the club.
The spokesperson said they would refer requests to the club's committee for comment.
The SPCA has also been approached for comment.
