No, you cannot apply for the SNQ quota if you have already taken a private seat in an engineering college through KCET .

The SNQ (Supernumerary Quota) is meant for students from economically weaker sections and is available only during the counselling . You have to choose SNQ at the time of option entry . Once you take a private seat, you are not eligible for SNQ.

So, if you want to apply under SNQ, you must select it during the option entry round , not after taking any seat.