Mr Quang died in a military hospital in Hanoi from a “serious illness, despite efforts by domestic and international doctors and professors,” Vietnam Television reported.
Rumours of Quang’s illness had been circulating on social media for months, but the details are as yet unconfirmed.
At one of his last appearances was during a visit by Indonesian President Joko Widodo to Hanoi on September 11.
Mr Quang caused concern by appearing visibly unwell, and stumbling as he stepped onto a platform to inspect a guard of honour.
Mr Quang was one of Vietnam’s top three leaders – Vietnam has no paramount ruler and is also led by the prime minister and Communist Party chief.
The role of the president is mostly ceremonial.
Mr Quang had been serving as president since April 2016.
Prior to his presidency, he served as Minister of Public Security, an organisation with broad powers and a remit that includes intelligence gathering and thwarting domestic and foreign threats to the party.
Originally from a small farming community 70 miles south of Hanoi, Quang rose through party ranks to become a police general and member of Vietnam’s powerful decision-making Politburo.
Bui Duc Phi, chairman of the village in which Quang was born, said: “We are saddened to hear the news that the president has died.”
Mr Quang had been withdrawing from public life over past months, raising speculation about his health.
His last public appearance was at a Politburo meeting of the ruling Communist Party and a reception for a Chinese delegation on Wednesday.
Mr Quang hosted President Donald Trump during his first state visit to the communist country last year.
Adam Sitkoff, executive director of the American Chamber of Commerce in Hanoi, said: “He was the first Vietnam president to meet with us, speak at a US-Vietnam business summit.
“He was a big believer in the private sector and improving relations between the US and Vietnam.”
Mr Sitkoff added that the death was not expected to alter the government’s economic policies.