If I pay off a judgment in full, mustI also pay interest on the amount?

In all states in this country, if a judgment is entered against a judgment creditor, besides being awarded the costs of the litigation supporting the judgment and the judgment, the judgment creditor also is entitled to accrued interet on the judgment's total running from the date of its entry until paid off.

In most states, the interest on an established judgment is set by statute. Typical interest rate is ten percent per annum.

Although you did not verbally agree to pay the interest on the judgment nor is it listed in the judgment, you are obligated to pay accrued interest on the judgment by your state's statutes on the subject. The rationale is that if a judgment debtor had an interest free judgment, there would be no incentive to pay it off.

You may well have to pay the interest, if it is post-judgment interest. Once there is a judgment against you, it effectively has to be paid immediately. If it is not, the judgment creditor (the party who sued you and won) may also recover "post-judgment" interest at  statutory interest rate. This post-judgment interest has nothing to do with the agreement; it's something the creditor may seek under law. On the other hand, if you did pay promptly and this is not post-judgment  interest, you would seem to not have to pay it; the creditor may not unilaterally change the judgment to seek more money from you, even if the creditor probably would have been entitled to it had he sought it during the litigation. Another factor, though, to consider, is whether it is worthwhile--e.g. cost effective; worth the distraction and aggravation--to fight this interst, or whether, all in all, you'd be better served by paying it.