expression, ichidan verb
pay one's bill; settle one's account
Used when paying a bill at a restaurant, bar, or similar establishment. Often implies finishing the payment and leaving.
食事が終わったら、勘定を済ませて店を出た。
After finishing the meal, I paid the bill and left the restaurant.
勘定を払う simply means 'to pay the bill', while 勘定を済ませる emphasizes completing the payment process, often with the nuance of settling up and leaving.
Fixed expression composed of 勘定 (bill, account) + を + 済ませる (to finish, to settle).