Police said the trio were on their way to check a dam when the quad bike hit a ledge and flipped.
Jack died at the scene. His father and brother, Charlie, suffered minor injuries.
The family has strong links to New Zealand, where Mr Moir is from and where Jack was born.
Jack's mother, Louise Moir, said the family had received huge support from friends and extended family members, who were travelling from afar to be with them.
"We've just got to get our heads around things," she said.
"Noel's parents are coming over from New Zealand ... and my mum and dad are coming over from England.
"We've had lots of friends over and we're getting lots of support."
Inspector Guy Guiana told the Herald police were investigating the incident and whether any charges would be laid was not yet known.
"It's a bit early to tell at this stage. Because of the nature of the accident, it's being investigated by our crash and investigation unit.
"Obviously it's a very sensitive issue with the family, so [we] wouldn't be pursuing the investigation until the little chap's buried."
Mr Guiana said police in the rural area continued to push safety messages regarding quad bikes and similar vehicles, given the popularity of them among locals.
"We certainly put out a message [at the weekend] asking people to seriously consider the use of safety devices such as roll-bars and safety helmets, because we have a lot of accidents on quad bikes."
A Facebook page called Support for Moir Family had been created.
Family friend Christine Pinkerton expressed her sadness on the site.
"The past few days has shown me how unpredictable life can be ... I'm gonna be holding my baby a lot closer from now on and telling my husband I love him every day before he goes to work ... It's just too precious."
Belinda Mahony said: "He was a very loved little boy."