Fujifilm XQ1 has a 12.0MP 2/3" (8.8 x 6.6 mm ) sized CMOS X-TRANS II sensor . On the other hand, Fujifilm X-T1 IR has a 16.0MP APS-C (23.6 x 15.6 mm ) sized CMOS X-TRANS II sensor and features EXR Processor II processor.
Another difference between these two cameras is that Fujifilm X-T1 IR's sensor lacks 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 XQ1 and X-T1 IR sensor size comparison.
As seen above, Fujifilm X-T1 IR has a 6.3x Larger sensor area than Fujifilm XQ1. 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.