= Solution
{c}
Conjugate by a <projective transformation> sending $P$ to $[1:0]$ and $Q$ to $[0:1]$. A matrix representing the conjugated transformation must have zero diagonal, since its first column represents $[0:1]$ and its second represents $[1:0]$. Therefore it has the form
$$
A=\begin{pmatrix}0&a\\b&0\end{pmatrix},\qquad ab\ne0.
$$
Its square is $abI$, a nonzero scalar matrix, which induces the identity on the <projective line>. Conjugating back proves
$$
\boxed{\tau^2=\operatorname{id}.}
$$
For the requested explicit <involution>, take
$$
\boxed{\tau([x:y])=[y:x].}
$$
More generally $[x:y]\mapsto[ay:bx]$ works whenever $ab\ne0$.
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