There were unconfirmed reports that two villages in the country's far west were covered with water, Julian McLeod of the Solomon Islands National Disaster Management Office said.
"Two villages were reported to have been completely inundated," McLeod told Australian Broadcasting Corp radio.
"We have received reports of four people missing."
Many geological agencies, including those in Australia and Japan, put the magnitude of the quake at 8.1 but the US Geological Survey's initial estimate was 7.6.
The initial tremor appeared to have immediately been followed by a second one of similar magnitude, Japan's meteorological agency's website said.
The US centre said the quake – which reportedly caused buildings to shake as far away as Japan - struck about 10km beneath the sea floor about 350km northwest of the Solomons capital of Honiara at 7:39am on Monday (2039 GMT on Sunday).
The Hawaii-based Pacific Tsunami Warning Centre said an earthquake of that strength could cause a destructive tsunami and issued a warning for the Solomon Islands and neighbouring Papua New Guinea.
It ordered a lower-level "tsunami watch" for other places, including most South Pacific countries.
The centre said it was monitoring the situation and that Hawaii's status could be elevated to warning or watch.
Abiah said police in Gizo had been warning residents to move to higher ground away from the coast when the tsunami hit, but communication links were lost soon afterwards.
"We have lost radio contact with the two police stations down there and we're not getting any clear picture from down there," he said.
Australia issued a tsunami alert for the far north coast of Queensland state, including the tourist city of Cooktown and the Willis and Barrier Reef islands.
Residents were warned to get out of the water and stay away from low-lying areas.