= Solution
Use independent percolation configurations in $H$ and in its reflected copy $H'=(-\infty,-2]\times\mathbb Z^{d-1}$. Require event $A$ in both copies and close the intervening site $(-1,0,\ldots,0)$. This has probability $(1-p)Q^2$. Because either distinguished infinite cluster meets its boundary hyperplane only at its distinguished origin, the closed intervening site prevents it from leaving its own half-space. The resulting whole-space configuration therefore has two distinct infinite clusters. Whole-space supercritical Bernoulli percolation has an almost surely unique infinite cluster, so $(1-p)Q^2=0$ and hence $Q=0$.
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