Consider a supplement $T'$ to $T$, so that $T \times T'$ is a maximal torus of $G$.  The centraliser $H_0$ of $T'$ in $G$ is then a reductive group, defined over $Q$ if $T'$ is, whose maximal subgroup is $T \times T'$. Taking $H$ the derived group of $H_0$ will kill the central $T'$ and leave you with a group having $T$ as maximal torus — it is not centralised as you can check by constructing an appropriate $\mathfrak{sl}_2$-triple.