Consider a supplement $T'$ to $T$, so that $T \times T'$ is a maximal torus of $G$.  The centraliser $H$ of $T'$ in $G$ is then a reductive group, defined over $Q$ if $T'$ is, and $T$ must not be in the centre of $H$ — and will actually never be as you can check by constructing an appropriate $\mathfrak{sl}_2$-triple.