Yes.
Certainly we do in Australia. In 1999, there was a referendum to decide whether or not Australia would abandon the Monarchy and become a Republic. It failed in every single state.
There is a reasonably large movement that would like Australia to become a republic, but even most of them greatly respect the current Queen.
Most Australian conservatives (though certainly not all) are Monarchists - I most certainly am.
But it is important to understand that she is the Queen of Australia - not the Queen of England or Great Britain - when she is acting as our Monarch. It's the same person, of course, but legally it's different and the distinction is important. Under the Statute of Westminster 1931 (ratified in Australia in 1941), Australia is a fully equal member of the Commonwealth of Nations with absolutely equal status to Britain (Canada and New Zealand also have this status - Ireland and South Africa had it until they chose to become republics). Australians have as much say over the future of the Monarchy as Britons do, under the law, and that is why it is accepted. Prior to 1941, Britain still handled our foreign affairs, and that would no longer be acceptable.
She is, as far as we are concerned, Her Majesty Elizabeth the Second, by the Grace of God Queen of Australia and Her other Realms and Territories, Head of the Commonwealth.
The Monarchy is still important here - when we had our awful bushfires last year (the worst natural disaster in Australia's history in terms of lives lost), HRH Princess Anne, the Princess Royal, was immediately dispatched to represent the Queen and the Royal Family at Australian memorials. Among his many titles, Prince Charles is a Knight of the Order of Australia, Prince Phillip holds the rank of Field Marshall of the Australian Army and Admiral of the Fleet of the Royal Australian Navy.
The last time the Queen visited Australia was 2006, to open the Commonwealth Games in Melbourne. There was considerable controversy when it was discovered that organisers had omitted 'God Save The Queen' from the opening ceremony. They were forced to reverse that decision in the face of public outcry and I was there in the crowd during the ceremony. Everybody in that stadium of 100,000 people came to their feet as it began.
Oh, yes, we acknowledge her. She is our Queen.