But for the master equation to be valid, \\(\[H_{i,j}\]\\) can't be any old infinite square matrix.

There are two criteria to be satisfied.

Look at the matrix multiplication involved in computing \\(H(\sigma)\\), for a general stochastic state \\(\sigma\\). This is an infinite sum, which is required to converge for \\(H(\sigma)\\) to be well defined.

A sufficient condition to guarantee this convergence is that in the graph for the Markov chain, each node has only a finite number of neighbors. From this it follows that the rows (and columns) will only have a finite number of nonzero entries, which makes \\(H(\sigma)\\) well-defined.