Mr Johnson is meeting USA secretary of state John Kerry in a get-together which London hopes will demonstrate that Washington still values the "special relationship" between the two countries after the surprise vote to withdraw from the EU.
Appointed last week, Johnson was in Brussels for his first Foreign Affairs Council event, where he said future UK-EU co-operation on foreign policy and security were discussed as well as the failed coup in Turkey and the truck attack in Nice.
British Foreign Secretary Boris Johnson (L, front) and Dutch Foreign Minister Bert Koenders (R, front) arrive at an European Union foreign ministers' meeting at its headquarters in Brussels, Belgium, July 18, 2016.
"You've accused the the current U.S. president Barack Obama of harboring a part-Kenyan's, quote, 'ancestral dislike for the British Empire, ' unquote, while claiming, I think untruthfully at the time, that he didn't want a Churchill bust in the White House".
Britain's parliament was due to vote later on Monday on replacing the country's ageing Trident nuclear deterrent, a massive commitment costing £41 billion (Dh200bn).
The former Mayor of London, who was named as Prime Minister Theresa May's Foreign Secretary last week, will share 17th-century mansion Chevening, in Kent, with Brexit Secretary Mr Davis and International Trade Secretary Dr Fox.
He will then host a meeting of counterparts from Germany, France and Italy, and the EU High Representative to discuss the conflict in Syria.
"We are talking about people now, so all sorts of statements and acts should be firmly anchored in the law", he said.
"There are lots of things to work on with Britain".
A spokesman for Johnson said "it would not be appropriate for him to continue writing his long-standing column for the newspaper given his new role".
Britain's Boris Johnson, who made his name as a Brussels-bashing journalist in the 1990s, was determined to avoid making headlines when he returned to his old stamping ground on Monday.
His critics accuse him of exaggerating his stories to play to the eurosceptic gallery at home.
Officials in Brussels stressed they would welcome Mr Johnson but there is little doubt his Brexit role ruffled feathers.
She refused to be drawn on Britain's negotiations for its departure from the bloc, which European Union leaders insist can only start once London invokes Article 50 of the EU's Lisbon Treaty to trigger the divorce.