Can a hereditary graph class contain almost all, but not all, n-vertex graphs?
The answer is no - for a fixed $Q$ let $t$ be the number of vertices in the smallest graph $H$ not in $Q$. Now, consider $n$ much bigger than $t$. For a random graph on $n$ vertices, the probability that the $t$ first vertices induce $H$ depends only on $t$. Partitioning the vertex set into $n/t$ disjoint sets of size $t$ and considering the probability that none of the sets are equal to $H$ shows that the probability of being in $Q$ tends to $0$ as $n$ increases.