If you give a client a refund via check for something they already paid, do you need to post the credit to their account?

No, not when you are issuing a cash (including a check) refund. If they already paid, their account is in order if you issue a cash refund. You would dr refunds, sales returns, or some such account (depends on your business) and credit cash.

Let me use numbers to make this more clear. If they bought something for $20 in one month and paid for it in the next month, you had a credit to payables in the first month and a debit in the second month. These zero each other out. Their account is correct.

That makes perfect sense. Thank you for the example, too!