The act is significant as the garden, in Dublin, is dedicated to people who fought for Irish independence.
Dozens of protesters gathered nearby amid one of the biggest ever security operations in the country.
Earlier, the Queen was welcomed by President Mary McAleese after flying into Baldonnel military airbase.
Earlier it emerged that a pipe bomb found on a bus bound for Dublin on Monday had been made safe by an Irish army team.
Upon landing at the military airbase, just outside Dublin, the Queen emerged wearing an outfit of symbolic emerald green.
The monarch later emerged from dinner wearing a white outfit.
God Save The Queen was played when she arrived at the Garden of Remembrance to lay the wreath - regarded as a highly symbolic act as the garden commemorates Irish people who fought over the centuries against British rule.
A one-minute silence followed the laying of the wreath, after which the Irish Tricolour was raised from half to full mast and the country's national anthem was played.
Meanwhile, riot police officers jostled with demonstrators at two separate protests on streets several hundred yards from the garden.
National anthems
The sounds of protesters could be heard during the laying of the wreath and black balloons were released by some demonstrators.
The Queen then went on to Trinity College to be shown the Book of Kells, an ancient illuminated manuscript in Latin, containing the four gospels of the New Testament.
The monarch will then go on to a reception to meet Trinity scholars, musicians and artists.
Continue reading the main story
Mark Simpson BBC Ireland Correspondent The Queen, dressed in Irish green, finally set foot on Irish soil at 1203 BST.
It was one small step for the Queen - one huge moment in British-Irish history.
It is 100 years since a British monarch has been in Ireland. Many thought a king or queen would never be welcomed back.
Instead, Ireland rolled out the red carpet for her nearest neighbour.
Whatever the security concerns, this has been a landmark moment in Dublin. In truth, it's one that most people living in Ireland today thought they would never see.
At Baldonnel airbase, the royal party was greeted by an Irish Air Corps guard of honour and presented with flowers by a south Dublin schoolgirl, eight-year-old Rachel Fox.The Queen and Duke of Edinburgh were then driven in a bombproof and bulletproof car, with a 33-motorcycle escort, to Aras an Uachtarain - the official residence of the Irish president in Dublin's Phoenix Park. The visit is taking place following Mrs McAleese's invitation.
The Queen shook hands with the president at the front of the residence before moving inside to meet the Taoiseach (prime minister) Enda Kenny.
The Irish flag and the union jack flew side by side at the gates to the building, where both countries' national anthems were played.
A 21-gun salute and an Air Corps flypast also marked the Queen's arrival.
She and Prince Philip signed the visitors' book in the residence's lavish state ballroom, before having lunch with the president and her husband.
King George V was the last monarch to visit the country, in 1911, when what is now the Republic was then part of the UK.
Ahead of the visit, UK Prime Minister David Cameron said: "One hundred years on from the last time a British monarch visited Ireland, I think there is a great sense of history and occasion."
He added: "I think the real effect... will be a marker that just as we are solving some of the problems there have been between us in the past, just as we are helping each other through these difficult economic times, now is a great moment for people in Britain and people in Ireland to remember what it is we share."
Mr Cameron will join the Queen and Duke of Edinburgh for part of their trip on Wednesday, while Foreign Secretary William Hague is following the usual practice of accompanying the royals throughout their visit.
The president and the Queen, who have met a number of times, chatted animatedly There were scuffles between police and republicans protesting against the visit in the centre of Dublin at Tuesday lunchtime.
Up to 200 supporters of the Eirigi socialist republican party later retreated to stage a rally nearby.
The Queen will attend events at Trinity College Dublin, the National War Memorial Gardens in Islandbridge and Croke Park stadium.
Croke Park is the home of Gaelic games where in 1920, during the Irish War of Independence, British forces fired into the crowd at a football match, killing 14 spectators and players.
The Queen is also to make a speech at a state dinner at Dublin Castle.
There are also plans for the Queen and Prince Philip to visit the Irish National Stud in County Kildare, as well as the Rock of Cashel in County Tipperary and a technology park in Cork.
Controlled explosionThe bomb on the bus was found in a holdall in the luggage compartment on Monday night during a check in Maynooth, County Kildare, to the west of Dublin.
Continue reading the main storyThe Irish president's dressed in pink. No clash. Must have checked in advance”End Quote Peter Hunt Royal correspondent, BBC News About 30 people who were on board the bus were taken off and transported to Dublin in another vehicle.
The device was later made safe in a controlled explosion carried out by an Irish army bomb disposal team.
A coded bomb threat relating to London had been received on Sunday, the first issued by Irish dissidents outside Northern Ireland in 10 years, officials said.
However, the threat level for Northern Ireland-related terrorism in Britain remains unchanged at substantial. In Northern Ireland it is severe.
The cost of the security operation has been estimated at 30 million euros (?26m), with measures including:
Deployment of more than 6,000 Irish police and Defence Forces personnel onto the streets of DublinIncreased surveillance of known republican dissidents A ring of steel, comprising 25 miles of crowd-control barriers, installed around the Irish capitalChecking of thousands of manhole covers and lamp-posts, and parking bans imposed on 30 city centre streets
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