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I encountered notions of block length and error rate for quantum error correcting codes, which literature seems to just assume. Can someone please give precise definitions for these?

Usually we denote a quantum code by parameters $[[n,k,d]]$. Where $n$ is the number of physcial qubits used, $k$ - number of encoded (or logical) qubits, and $d$ the code distance.

Besides the definitions, how are block length and error rate related to those parameters $n,k$ and $d$?

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The block length is defined as the number of physical qubits you are using to encode the logical qubit. Thus in the parameters $[n,k,d]$ the value $n$ is your block length.

You can see this paper here for confirmation on what I just stated.

The error rate here is probably the error rate of the qubit.

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