  In this article, we study the set of parameters $c \in \mathbb{C}$ for which
two given complex numbers $a$ and $b$ are simultaneously preperiodic for the
quadratic polynomial $f_{c}(z) = z^{2} +c$. Combining complex-analytic and
arithmetic arguments, Baker and DeMarco showed that this set of parameters is
infinite if and only if $a^{2} = b^{2}$. Recently, Buff answered a question of
theirs, proving that the set of parameters $c \in \mathbb{C}$ for which both
$0$ and $1$ are preperiodic for $f_{c}$ is equal to $\lbrace -2, -1, 0
\rbrace$. Following his approach, we complete the description of these sets
when $a$ and $b$ are two given integers with $\lvert a \rvert \neq \lvert b
\rvert$.
