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Web design involves writing HTML for structured content and CSS for its presentation. Now, if you're talking about writing a web based application... Yes, you may use C++ to write the server side backend which would generate your pages. But it is not recommended as C++ is not suitable for sandboxes environment and posses a major security risk. Just imagine if you open a website and it contains a c++ program with major memory leaks.
At the very least, you need to use HTML and CSS to construct a very basic website. However, I presume you mean create a dynamic web application for your website, too. Common languages used for this purpose include PHP, Java, and Python. You can use C++, as well. There are several good C++ web frameworks or libraries.
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