After the playing of the two countries' national anthems and a review of the Guard of Honour, the royal and presidential couples headed for Josipovic's office for talks.
Queen Margrethe and President Josipovic exchanged the highest state decorations. The Danish queen was honoured with the Croatian Order of King Tomislav, while the Croatian president was awarded the Danish Order of the Elephant.
During the day, Queen Margrethe II and her husband will also meet Parliament Speaker Josip Leko. They will wrap up the first day of their visit by attending a state dinner at Josipovic's office.
This is the Danish royal couple's first official visit to Croatia since the country gained independence. They visited Croatia together only once, in 1978, when they stayed on the Adriatic coast.
The delegation accompanying the royal couple consists of representatives of some 30 Danish companies, as well as the minister of health, agriculture and fisheries.
On the second day of their visit on Wednesday, the royal couple will attend a Danish-Croatian business forum, and Queen Margrethe II will attend, alongside President Josipovic, a regional forum on diabetes.
The Danish royal couple is expected to attend two cultural events on Wednesday, the Croatian premiere of "Why Me?", a co-production by the Croatian theatre Mala Scena, the Danish children's theatre company Passpartout and Sweden's Landskron theatre, and the screening of the Croatian-Danish co-production "Itsi-Bitsi" at the Zagreb Film Festival.
The high guests will also visit Zagreb's St. Mark's Church, the Glyptotheque gallery of the Croatian Academy of Sciences and Arts, where Danish architecture between 1945 and 1975 is being presented, and the Museum of Ethnography.
Queen Margrethe II and Prince Henrik will also visit the northern town of Koprivnica and the Danish-owned companies Carlsberg and Hartmann operating there, as well as the southern coastal city of Split.