The quake hit at a depth of 60.5 km and was followed by aftershocks of a magnitude of 5.0 to 5.1.
An earlier report from the USGS gave the magnitude of the quake at 7.0 and the depth of 10km.
Samud Sergi, an official from the search and rescue agency in Ternate, said the quakes had not been felt in the city.
A spokesperson for the emergency disaster agency in Tohomon for North Sulawesi said "they had felt nothing in the area and had heard of no damage or casualties".
The city of Ternate has a population of around 200,000.
The Pacific Tsunami Warning Center said in a statement that based on the available data, there was no current tsunami threat from the event.