Often the developers of these sites have implemented an API for this purpose, using SOAP, JSON or some similar protocol. The site you give has such an API: http://www.xe.com/xecurrencydata/, but it isn't free. There are some free ones out there.
Qt has classes for implementing JSON-based protocol. SOAP is XML-based, and Qt has XML classes too. And classes for HTTP communication with servers. Look at the examples and tutorials that came with your Qt installation.
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