Hi. No, the O-ICK1 and any other similar "gel-stick" cleaning tools are used to lift dust and debris off the sensor. To remove skin oils, you'll need to wet clean the sensor using dedicated sensor swabs and sensor cleaning fluid.
Whilst I've wet cleaned DSLRs on numerous occasions, I've never done so with the Q. The main issue will be finding swabs that are small enough for the Q's tiny sensor. I'm sure they must exist though... I'm just going offline for the evening, but if no-one else has pointed you in the right direction by tomorrow morning, I'll see what's available and let you know