The quake hit 154 kilometers (96 miles) southwest of Dadali and 170 kilometers west of the capital Honiara, according to the US Geological Survey.
USGS had initially measured the earthquake at 7.0 magnitude but revised it down.
It was followed by a 5.3 tremor.
"Based on all available data a destructive Pacific-wide tsunami is not expected," the Pacific Tsunami Warning Centre said.
The Solomons are part of the Pacific "Ring of Fire", a zone of tectonic activity known for its frequent earthquakes and volcanic eruptions.
Also Read
"It is unlikely if there will be much damage, if any."
A spokesman for the National Disaster Management Office in the Solomons said that "at the moment, no reports have come in of any damage".
In 2013, the Solomons were hit by a tsunami after an 8.0-magnitude quake, leaving at least 10 people dead and thousands homeless after buildings were destroyed.