Both Casio EX-G1 and Fujifilm XQ1 have 12.0 MP resolution sensors but Casio EX-G1's sensor is 1/2.3-inch (6.17 x 4.55 mm ) and Fujifilm XQ1's sensor is 2/3-inch (8.8 x 6.6 mm ).
Since Fujifilm XQ1's has a larger sensor area with the same resolution, this means that it also has a larger pixel area hence better light collecting capacity for a given aperture compared to Casio EX-G1.
Below you can see the EX-G1 and XQ1 sensor size comparison.
As seen above, Fujifilm XQ1 has a 2.1x Larger sensor area than Casio EX-G1. 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.