Nikon 1 J3 has a 14.0MP 1" (13.2 x 8.8 mm ) sized CMOS sensor . On the other hand, Sony A7R III has a 42.0MP Full frame (35.9 x 24 mm ) sized BSI-CMOS sensor and features Bionz X processor.
Sony A7R III's sensor provides 28MP more than Nikon 1 J3's sensor, which gives a significant advantage in real life. You can print your images larger or crop more freely.
On the other hand, please keep in mind that Max sensor resolution is not the only determinant of resolving power. Factors such as the optical elements, low pass filter, pixel size and sensor technology also affects the final resolution of the captured image.
One other difference between these two cameras that is worth mentioning is that Sony A7R III's sensor doesn't have an anti-alias (Low-Pass) filter. Removing anti-alias filter increases the sharpness and level of detail but at the same time, it increases the chance of moire occurring in certain scenes.
Below you can see the 1 J3 and A7R III sensor size comparison.
As seen above, Sony A7R III has a 7.4x Larger sensor area than Nikon 1 J3. Larger sensors give photographer more control on the depth of field and blurry background compared to smaller sensor when shot in same focal length and aperture.
DxOMark Sensor Scores
DxOMark is a benchmark that scientifically assesses image quality of camera sensors. It scores camera sensors for color depth (DXO Portrait), dynamic range (DXO Landscape) and low-light sensitivity (DXO Sports), and also gives them an overall score. Nikon 1 J3 and Sony A7R III sensors have been tested by DxO and the results show that A7R III has a better overall score of 100, 48 points higher compared to 1 J3's score of 52.