Let $\mathbf{M}$ be the mean matrix of a multi type branching process $(\mathbf{Z}^{(n)})_{n\geq1}=((Z^{(n)}_1,\ldots,Z^{(n)}_k))_{n\geq1}$. This matrix is defined as follows $$M_{i,j}=\mathbb E\left(Z^{(1)}_j\,\big|\,\mathbf{Z}^{(0)}=\mathbf{e}_i\right)$$ Let $\mathbf{Q}$ be the transition matrix of the process as a Markov chain. The entries of $\mathbf{Q}$ are indexed by $\mathbb{N}^k\times\mathbb{N}^k$, since a state of the process is of the form $(z_1,\ldots,z_k)\in\mathbb N^k$, where $\mathbb N=\{0,1,2,\ldots\}$.
Is there any known algebraic relation between these two matrices?
No nonambiguous one, and for a good reason: there is much more information in the latter than in the former.