 Solution

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 formIts square is $abI$, a nonzero scalar matrix, which induces the identity on the projective line. Conjugating back provesFor the requested explicit involution, takeMore generally $[x:y]\mapsto [ay:bx]$ works whenever $ab\ne 0$.