By definition, a function $u$ is quasi-convex if, for every $a\leqslant c\leqslant b$, $u(c)\leqslant\max(u(a),u(b))$, $u$ is quasi-concave if, for every $a\leqslant c\leqslant b$, $u(c)\geqslant\min(u(a),u(b))$, and $u$ is quasi-linear if $u$ is quasi-convex and quasi-concave.
Note that $\min(a,0)+\min(-a,0)=-|a|$ and $a\mapsto-|a|$ is not quasi-convex. Hence, the function $f$ defined by $f(a)=\mathrm E(\min(aX,Y))$ for some random variables $X$ and $Y$, is not quasi-linear in general.
On the other hand...
If $X\geqslant0$ with full probability, since each function $a\mapsto\min(ax,y)$ with $x\geqslant0$ is nondecreasing, $f$ is nondecreasing. Likewise, if $X\leqslant0$ with full probability, since each function $a\mapsto\min(ax,y)$ with $x\leqslant0$ is nonincreasing, $f$ is nonincreasing. In both cases, $f$ is quasi-linear.