"We can be informed by many sources of the ills of our sociey; it is important that we are reminded of its goodness," he told the Senate chamber.
Mr Hurley said he will also do his best to highlight the importance of the work of governor-generals in "depicting a nation to its people".
"Australia can be provided with a reflection of themselves and other countries," he said, paying tribute to the strength of Australia's democratic system.
"My emphasis on community engagement will sit alongside the critical constitutional role performed by the Governor-General. Australia has an enviable record as a parliamentary democracy."
Support for veterans, asylum seekers, Indigenous Australians
He said his time as NSW governor has made him acutely aware of the "richness" and cultural diversity of Australia.
"I have seen this richness of spirit at work in the people and the organisations assisting our veterans, our rural communities under stress, asylum seekers who we have welcomed into our country and our Indigenous brothers and sisters.
"There are countless organisations and people demonstrating how rich this country is every day.
"As Governor-General, I am committed absolutely to supporting these people in their work."
Mr Hurley also drew on a quote by Australian author David Malouf that is etched on a glass plaque at Sydney's Circular Quay.
Malouf, he noted, is representative of the diversity of Australia, having a Lebanese father and British mother of Portuguese decent.
"Australia is still revealing itself to us. We oughtn't to close off possibilities by declaring too early what we have already become," the plaque reads.
"Australia is not a finished product," Mr Hurley stressed.
Decades of service in the army
Mr Hurley became the NSW governor in 2014 after 42 years of service in the Australian Army, the last three of which he spent as chief of the defence force.
Accompanied by his wife Linda, Mr Hurley received the general salute in the parliament forecourt.
In the Senate chamber, Australia's Chief Justice Susan Kiefel swore in the 27th governor-general, reading and signing the oaths of allegiance and office.
Prime Minister Scott Morrison and Mr Hurley then signed the proclamation to complete the ceremony.
The governor-general opened his acceptance speech with an acknowledgement of country in Ngunnawal language, after calling for Indigenous languages to be taught more widely in schools in an Australia Day speech.
In December, he urged people to look beyond his military background after criticism that three of the last four appointments of governor-general have been retired army generals.
'Understanding people's challenges'
Mr Morrison gave some insight into his choice after appointing him late last year.
"General Hurley is known for looking people straight in the eye. Not up and not down. He was that way with those he led in the military and he's been that way as a governor and throughout his life," Mr Morrison said in December.
"Looking eye-to-eye, face-to-face, understanding people's challenges and issues one-on-one, in a very direct and very humble and a very humane way."
Facing questions about why he did not consider a woman to fill the role, Mr Morrison said Mr Hurley's selection was a "package deal" with his wife of more than 40 years.
The governor-general is appointed by the Queen in Commonwealth countries, on the advice of the prime minister, to be the representative of the Crown.
Mr Hurley was awarded the Distinguished Service Cross for his leadership during Operation SOLACE in Somalia in 1993 and was appointed a Companion of the Order of Australia in 2010 for eminent service to the army.