Tremors were felt in the entire northeast region, including the neighbouring state of Meghalaya. The aftershock was also felt in the state of West Bengal.
The agency said aftershocks could be expected, but the earthquake, which struck at a depth of 112 km (69.6 miles), was unlikely to cause damage.
The US Geological Survey said the magnitude 6.0 quake struck off the island's southern coast at 2pm on Saturday. It was centered 45 kilometres (28 miles) south of Sumberpucung town of Malang District in East Java province, at a depth of 82 kilometres (51 miles).
Two minor earthquakes were felt in Fujairah early on Monday morning, according to the National Seismic Network of the National Centre of Meteorology (NCM).
The quake's epicentre was near the rural town of Mansfield in the state of Victoria, about 200 km (124 miles) northeast of Melbourne, and was at a depth of 10km (six miles). An aftershock was rated 4.0.
According to the epicentre the temblor was 37.1 kilometres north-northeast of Hualien County Hall at a focal depth of 45 km. No injuries or damage from the quake had been reported at the time of publication.
The US Geological Survey says the first quake had a magnitude of 6.1 and struck about 17 miles south of Naalehu. The agency says a 6.2 magnitude Earthquake struck about 20 minutes later in the same area.
The quake struck at around 2:46am local time (1746 GMT). No damage was reported at a nuclear fuel preprocessing plant in the area, public broadcaster NHK said.
The quake was at a depth of 13 kilometres (7.5 miles), about 180 kilometres northeast of Tual city, Indonesia's easternmost Papua region, according to the United States Geological Survey said, but there was no tsunami warning or immediate reports of damage.