  Let $\perp$ be a unitary polarity of a finite projective plane $\pi$ of order
$q^2$. The unitary polarity graph is the graph with vertex set the points of
$\pi$ where two vertices $x$ and $y$ are adjacent if $x \in y^\perp$. We show
that a triangle-free induced subgraph of the unitary polarity graph of an
arbitrary projective plane has at most $(q^4+q)/2$ vertices. When $\pi$ is the
Desarguesian projective plane $\mathrm{PG}(2,q^2)$ and $q$ is even, we show
that the upper bound is asymptotically sharp, by providing an example on
$q^4/2$ vertices. Finally, the case when $\pi$ is the Figueroa plane is
discussed.
