Facebook and other firms have been criticised for failing to block a live-stream of the attack.
In a letter to Japan's Prime Minister Shinzo Abe, who chairs the G20, Mr Morrison asked for leaders to discuss the issue at the upcoming G20 meeting.
On Friday, twin shootings at mosques in Christchurch killed 50 people.
The gunman filmed the attack and the live-stream on Facebook lasted for 17 minutes.
Despite the original video being taken down, it was quickly replicated and shared widely on other platforms, including YouTube and Twitter.
