Trieste is most likely a feminine variation of Tristan.
As a Christian given name, Salome has been in occasional use since the Protestant Reformation. This was due to a second person of this name in the New Testament: one of the women who witnessed the crucifixion and later discovered that Jesus' tomb was empty. It is used in Georgia due to the 4th-century Salome of Ujarma, who is considered a saint in the Georgian Church.
The Apollo Belvedere is a famous marble statue from classical antiquity, in which case it refers to the Belvedere Gallery of the Vatican Museum in Rome, where the sculpture once stood. The Apollo Belvedere is allegedly responsible for inspiring American painters John Trumbull and Gilbert Stuart's portrayals of George Washington. This may be the inspiration behind its use in America.
It was used by American cartoonist George Webster Crenshaw in his comic strip Belvedere, which ran from 1962 to 1995, where it belongs to a male hound dog.
Already an uncommon masculine name, it slowly grew in popularity for American girls beginning in 1978 after the start of the sitcom WKRP in Cincinnati, which featured a character with this name. Though it remained more common as a feminine name, it got a boost for boys in 1994 from another television character on the drama Party of Five. In the United Kingdom and Australia it has always been more popular for boys.