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Counter comes from Middle English countour, from Old French conteor, from conter ‘to count’, from Latin computō ‘to calculate’.

1. Latinc. 75 BCE–200 CEputus¹‘pure, clean’
2. Latinc. 75 BCE–200 CEputō¹‘to reckon’
3. Latinc. 75 BCE–200 CEcomputō²‘to calculate’
4. Old Frenchc. 842–1400conter²‘to count’
5. Old Frenchc. 842–1400conteor²
6. Middle Englishc. 1150–1500countour²‘counting-table; accountant’
7. Englishearliest quotation 1590counter²

1. ¹ Wiktionary, Latin computo
2. ² Wiktionary, English counter
3. Earliest quotation in Wiktionary’s entry: 1590, John Greenwood: “He remaynes prisonner in the Counter in Woodstrete in…”.