Yes, IIRC the queen personally signed the new constitutional act in Canada in 1982.
Just a few years ago, the Canadian GG had to make an interesting constitutional decision. The prime minister had decided that because the opposition was in disarray, he could figuratively take a shit in their hat and there was nothing they could do about it even though he had a minority. He proposed a bill to remove the per-vote subsidies to political parties (since the Conservatives were in the black and all the other parties were hurting for money…); to add insult to injury, he added the clause to remove the right for the civil service unions to sue over wage discrepancies by sexual discrimination, probably because he thought it would be hilarious if the liberal party had no choice but to vote for it.
Instead, the other 3 parties decided to get together, and ask the Governor General to dismiss the government and appoint them the government as a coalition. The PM in return demanded the GG prorogue parliament for 3 months instead of proceeding to a confidence vote. So the GG as representative of the queen had to decide how badly to stick her nose into politics.
She took the safest, least disruptive course, suspended parliament for 3 months, since the parliament had not yet voted no confidence. By then the coalition fell apart and there was no need to make any more decision about it.
But that is the role of the monarch in modern government. She has a tremendous amount of power, but it is basically the “nuclear option”. If she throws her weight around unwisely, the government of the day, likely with the support of much of the people, will remove that option. However, she can block whatever she thinks is unwise. Usually the advice that she would consider doing so, or even that she would announce that she was unhappy, is enough to make the government pay attention.
The queen does not tell prime ministers what to do, but it is an unwise prime minister that ignores what she has to say.
In Canada (and I imagine the ANZ and elsewhere) the governor general has even less moral authority, and the queen herself has less contact with the government. But - actions of a government are technically done in her name, so she would from time to time pass on her thoughts. It’s free advice, prime ministers are probably smart to listen and consider it.
Technically, if something is totally abhorrent, a queen or GG can refuse to sign the bill or even call an election - the nuclear option that must be used very very wisely.