= Solution
Differentiate the <parametrized surface>. The coefficients of the <first fundamental form> simplify to
$$
E=Q^2+\frac{v^2}{4},\qquad F=0,\qquad G=1.
$$
If $c=\sigma_u\times\sigma_v$, then $|c|^2=E$. Moreover
$$
\sigma_{uv}\mathbin{\cdot}c=1,
\qquad
\sigma_{vv}\mathbin{\cdot}c=0.
$$
Thus the second-fundamental-form coefficients satisfy $M=1/\sqrt E$ and $N=0$. The <Gaussian curvature> is consequently
$$
\boxed{K=\frac{LN-M^2}{EG-F^2}=-\frac1{E^2}=-\frac1{(v^2/4+Q^2)^2}.}
$$
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