For a 1D Ising model, the transfer matrix is ##2\times 2##. For a 2D Ising model on an ##N\times N## lattice, the transfer matrix is ##2^N\times 2^N##, and the thermodynamic limit requires infinitely large matrices, ##N \rightarrow \infty##! Nonetheless, this can be done and used to derive the exact solution of the 2D Ising model. Most books on statistical physics and critical phenomena should cover this. For a generalized (Potts-like) model with ##m## possible states for each lattice site, the transfer matrix will be ##m^N \times m^N##.
EDIT: I mis-read your first sentence as a question. All of the above above applies to a Potts-like model with discrete degrees of freedom. For something like the Heisenberg model (or 2D Potts/XY model), where each site can take an infinite number of values ##m\rightarrow \infty##, I imagine a transfer matrix solution would become intractable.