Was Australia called Australia in the 1500s?

No. The continent that later became known as Australia was unknown in the 1500s, and only theorised about. However, early sea voyagers referred to the theoretical continental mass in the southern hemisphere as Terra Australis Incognita, which literally translates to "the unknown land of the South". It was also variously known as Locac or Lucach, India meridional (South India), Java le Grande and Terra Australis by Mercator (and others) in the latter half of the 1500s.

Spaniard sailor Pedro Fernández de Quirós (in Portuguese Fernandes de Queirós), searched for this new land in 1606 while serving in the navy, and called it Austrialia del Espíritu Santo, literally meaning 'Austria of the Holy Spirit'. The word "Austrailia", slightly different from the current "Australia", was a mixture formed by "Austria" and "Austral" ("Southern"). In those times the current nations of Spain and Portugal were under the rule of the same crown. The word Austrialia was intended to be a compliment to King Phillip III of Spain who was a member of the House of Habsburg (Austria). Different variations on the name were used in many languages. There was, however, a German document dating back to 1545 describing a southern land mass as Australia. (note the link below for a reference to this document)

The first use of the word "Australia" in English was in 1625-the words "A note of Australia del Espiritu Santo, written by Master Hakluyt.

The first "discovery" of the continent was in 1606 by Dutchman Willem Jansz, who believed it to be part of New Guinea. Soon after the next Dutch explorer, Dirk Hartog, came ashore in Western Australia in 1616, that part of the continent became known as New Holland, although the word Australische (the Dutch version of southern) was used throughout the publication Generale Beschrijvinge van Indiendescribing Dutch voyages in the East Indies, to describe regions south of the East Indies. Whoever compiled the index, however, used Australia in the index rather than Australische.

Even when James Cook charted the eastern coast in 1770, he named the Eastern seaboard New South Wales.

The name 'Australia' was not officially given until 1824.
The British colonies hadn't settled into Australia until the late 1700's, so it wasn't given its name until then.