Hartshorne II Example 7.6.2:The restriction of a sheaf of modules to a fibre.

This statement (that $\mathscr L$ is not generated by global sections) is under the additional assumption that either $a < 0$ or $b < 0$. First of all yes, the restriction of a sheaf $F$ to a closed subscheme $f:Y \to X$ is the pullback $f^*(F)$. In general, $\mathscr L$ restricts to $\mathcal O_{\mathbb P^1}(b)$ when restricted to a fiber of the first projection, and to $\mathcal O_{\mathbb P^1}(a)$ when restricted to a fiber of the second projection.

So if either $a$ or $b$ is negative, there is a closed subscheme $Y\subset Q$ along which $\mathscr L|_Y$ has no global sections, hence $\mathscr L$ has no global sections over $Q$ and cannot be globally generated.