No. Suppose the contrary that there are such integers $a$, $b$ and $c$.
If one of them is even, say $a$, then $a^2bc + 2 \equiv 2 \pmod 4$, which contradicts the assumption that $a^2bc + 2$ is a perfect square. We may then assume that $a$, $b$ and $c$ are odd, so they are either $1$ or $3 \pmod 4$. It follows from the Pigeon-hole Principle that two of them are congruent modulo $4$. Relabel if necessary, we may assume that $a \equiv b \pmod 4$, so $abc^2 + 2 \equiv c^2 + 2 \equiv 1 + 2 \equiv 3 \pmod 4$, which violates the perfect square assumption. $\square$