Consider the following matrix $\mathbf{H}\in\mathbb{R}^{k \times (M1+M2+1)}$.
\begin{align*}
\mathbf{H} =
\left[
\begin{matrix}
\mathbf{h}_{-M1}^{} & \cdots & \mathbf{h}_{-1}^{} & \mathbf{h}_{0}^{} & \mathbf{h}_{1}^{} & \cdots & \mathbf{h}_{M2}^{}
\end{matrix}
\right]
\end{align*}
where $\mathbf{h}_{i}$ is the $i^{th}$ column of $\mathbf{H}$. Now I am defining two new matrics $\mathbf{H_{1}}$ and $\mathbf{H_{2}}$ as \begin{align*} \mathbf{H_{1}} &= \left[ \begin{matrix} \mathbf{h}_{-M1}^{} & \cdots & \mathbf{h}_{-1}^{} \end{matrix} \right] \\ \mathbf{H_{2}} &= \left[ \begin{matrix} \mathbf{h}_{1}^{} & \cdots & \mathbf{h}_{M2}^{} \end{matrix} \right] \end{align*}
Therefore,
\begin{align*}
\mathbf{H} =
\left[
\begin{matrix}
\mathbf{H_{1}} & \mathbf{h}_{0}^{} & \mathbf{H_{2}}
\end{matrix}
\right].
\end{align*}
and $\mathbf{H^{T}}$ can be written as
\begin{align*} \mathbf{H^{T}} = \left[ \begin{matrix} \mathbf{H_{1}^{T}} \\ \mathbf{h}_{0}^{T} \\ \mathbf{H_{2}^{T}} \end{matrix} \right] \end{align*}
Can I write $\mathbf{H^{}H^{T}}$
\begin{align*} \mathbf{^{}HH^{T}} &= \left[ \begin{matrix} \mathbf{H_{1}} & \mathbf{h}_{0}^{} & \mathbf{H_{2}} \end{matrix} \right] \left[ \begin{matrix} \mathbf{H_{1}^{T}} \\ \mathbf{h}_{0}^{T} \\ \mathbf{H_{2}^{T}} \end{matrix} \right]\\ &=\left[ \begin{matrix} \mathbf{H_{1}^{}H_{1}^{T}} + \mathbf{h}_{0}^{}\mathbf{h}_{0}^{T} + \mathbf{H_{2}^{}H_{2}^{T}} \end{matrix} \right]\tag{*} \end{align*}
in terms of $\mathbf{H_{1}^{}H_{1}^{T}}$ and $\mathbf{H_{2}^{}H_{2}^{T}}$.
Is (*) correct?
Yes this is correct! Note that the three matrices summed in your expression (*) are all of dimension $k$ by $k$.