The Queen has urged people to find “common ground” and to respect “different points of view”.
Commentators say the remarks will be seen as referring to the Brexit debate, with MPs due to vote on the PM’s deal for leaving the EU again next week.
MPs rejected the deal last week but the UK will leave on 29 March with no deal unless they can agree on a way forward.
BBC royal correspondent Nicholas Witchell said there was little doubt the Queen was “sending a message”.
“It is impossible to imagine that the head of state would use a construction of words such as this without it being appreciated that they would be seen as a reference to the current political debate,” he said.
Her words echoed the theme of her Christmas broadcast, our correspondent added.
In other Brexit developments:
EU officials are due to discuss their no-deal legislation, with some members pushing for more generous rights for the UK on road haulage and airlines
The new chief executive of Goldman Sachs says a “difficult” Brexit could affect investment in the UK, becoming the latest business leader to voice concerns. Airbus called no-deal a “disaster” on Thursday
Former Foreign Secretary David Miliband has called for another referendum, saying MPs should not wait until “everything else has failed”
Cabinet minister Amber Rudd said she was “committed” to avoiding a no-deal Brexit and refused to rule out resigning over it
The Queen, as head of state, remains neutral on political matters and does not usually express her views on contentious issues.
However, speaking at an event to mark the 100 years of Sandringham Women’s Institute in Norfolk, the Queen said: “The continued emphasis on patience, friendship, a strong community focus, and considering the needs of others, are as important today as they were when the group was founded all those years ago.
“Of course, every generation faces fresh challenges and opportunities.
“As we look for new answers in the modern age, I for one prefer the tried and tested recipes, like speaking well of each other and respecting different points of view; coming together to seek out the common ground; and never losing sight of the bigger picture.”
She said these approaches were “timeless, and I commend them to everyone”.
The Queen’s remarks come as Prime Minister Theresa May is due to return to the House of Commons with her revised plan for Brexit on 29 January, after her deal was rejected by MPs in a historic defeat on 15 January.
MPs have also proposed a number of amendments with alternative proposals, which include delaying the 29 March deadline for leaving the EU to avoid exiting before a deal has been agreed.
Many MPs fear leaving the EU without a formal withdrawal agreement would cause chaos at ports and disruption for businesses, although the option is supported by some Brexiteer MPs who believe this view is exaggerated.
Another amendment seeks to win over Brexiteers to Mrs May’s deal by putting an “expiry date” on the backstop, an “insurance policy” in the withdrawal deal intended to ensure that there will be no return to a visible border between Northern Ireland and the Irish Republic.
Both the UK and the EU believe that bringing back border checks could put the peace process at risk but Brexiteers fear the backstop could tie the UK to EU rules indefinitely without any say over them.
The Queen’s appeal for common ground touched on the same issues as her Christmas message, in which she urged people to treat others with respect “even with the most deeply held differences”.