Converting ADC output to fixed-point format
You don't have anything to do, because $\rm Qm.n$ is just an "interpretation" of an integer. The only thing that really matters is that you need to correctly apply a shift operation when multiplying numbers; or when adding/subtracting numbers with different values of $\rm (m, n)$ - You can think of the $\rm Qm.n$ representation as a "tag" you attach to number, a bit like physical units.

In your case, the output of your ADC is an integer between $-32768$ and $32767$. This number can be interpreted as a $\rm Q15$ fixed-point number between $-1.0$ and $0.99997$. Or as a $\rm Q2.14$ fixed-point number between $-4.0$ and $3.99987$. Or as a $\rm Q8.8$ number between $-256.0$ and $255.992$. You don't have to do any arithmetic operation.