Is there any exchange website that directly converts to a specific currency once a cryptocurrent payment is received? I wonder if there is some exchange service that I can use for my webshop (Woocommerce) that directly when a customer pays in the webshop, convert the cryptocurrency money to a specific currency like USD or Euro.
What you're looking for is a payment gateway service, and if they take settlement in fiat such as USD or Euro.

It appears Coinpayments.net works on Woocommerce and allows you to withdraw in fiat pairs. You could also look into GloBee, BitPay, Paybear and Coinbase payments, and compare what they offer to see if they suite your needs.