Ms Bryce spoke out in favour of gay marriage during her Boyer Lecture address last night, stating she hoped Australia might become a nation where "people are free to love and marry whom they choose".
She then added: "And where perhaps, my friends, one day, one young girl or boy may even grow up to be our nation's first head of state."
It is the first time a governor-general has publicly supported a republic while still serving as the Queen's representative.
The republican movement has applauded the comments, but the Governor-General has drawn fire from other quarters.
Liberal Senator Dean Smith says Ms Bryce had crossed the line by voicing opinions on two sensitive political issues.
Senator Smith says Ms Bryce is held in high regard by many people and he is disappointed she has decided to involve herself in a political debate.
"Last night's departure into current political events will come as a slap in the face to many, many Australians, and a significant breach of trust because she would know better than most that that central office is so integral to stay above the day to day political fray," he said.
"She has stepped across the line in inviting a commentary around two very, very sensitive issues."
The national convenor of Australians for Constitutional Monarchy, David Flint, says the comments are regrettable and a mistake.
"I think it's a great pity because the constitutional system requires that the Crown be above politics so that when the representative of the Crown - which is the Governor-General or the Governor - speaks, they shouldn't be talking about politics. It's just it goes against the position," he said.
"There are a number of people who are now going to wonder about her. There's this sense of division that she's created and the position is not intended to be divisive - it's intended to unite and be above politics.
"So we've got commentators everywhere on these issues, couldn't she have left them alone until she was out of the office?"
Prime Minister Tony Abbott says the Governor-General is entitled to express her personal opinion.
"It's more than appropriate for the Governor-General, approaching the end of her term, to express a personal view on a number of subjects, and that's what she was doing," he said.
However, Coalition MP Kelly O'Dwyer says the issue is not a priority for the Government.
"I'm somebody who believes that Australia should have an Australian head of state," she told Lateline.
"I believe that we're a mature democracy and that we can and should have an Australian head of state. However, I'm not sure that this is a number one priority issue right now."
"There are a lot of issues on the to do list and I'm not sure this up there in the top 10."
Republicans welcome Bryce's comments
Federal Labor MP Graham Perrett says most Australians believe a home-grown head of state would be just as capable as a foreign monarch.
"Since the Australia Act in the late 1980s, the reality is we've drifted, to all intents and purposes, a long way away from that British throne," he said.
"The idea that because you have some hereditary link to one person makes you better than anyone in Australia is anathema to people, you know, sensible Australians."
Greens leader Christine Milne also welcomed Ms Bryce's comments, saying they reflected aspirations held by many Australians.
"I have no doubt that even the Monarchy sometimes wonders why it has taken Australia so long to get to the point of becoming a republic," he said.
"I think Quentin Bryce is showing great leadership as Governor-General and she is such a good role model and for her to come out now and support the idea of Australia becoming a republic is very welcome."
Greens Senator Sarah Hanson-Young, meantime, defended Ms Bryce over her comments on same-sex marriage.
She disagreed with Senator Smith's suggestion that the Governor-General had crossed a line in giving her opinion on sensitive political issues.
"It should be an issue that is above politics because it is about love, not because it should be dragged through the political debates of different political parties," Ms Hanson-Young said.
Monarchy in Australia: where you stand
In the lead-up to the federal election, our Vote Compass tool measured Australian attitudes on major political and social policies.
With more than 1.4 million responses, the data showed support for a republic appeared to have waned since the 1999 referendum on the issue, with only 38 per cent in favour of cutting ties to the monarchy.
Another 20 per cent were neutral. The strongest support for change came from those over 55 years of age.
Vote Compass asked respondents for their view on the statement: "Australia should end the monarchy and become a republic."
Use our interactive charts to explore the response.