Yes. For the most part, this would mean that you no longer owe the person money. So, you already paid them.
One example that I can think of that would break this pattern is if a person forgets that you owe him money, and you try to pay. He might say, "What's this for?" And you would reply, "Remember? I owed you money." You could also say, "I owe you money." But you will hear both.