There is no concept of "default file to include" in PHP. However, you can easily create a helper function to accomplish this. (Whether this makes sense or not, I leave for you to decide.)
function import(string $dir) {
include $dir . 'index.php';
}
Note that if your file to import contains variables, they will be "lost" in the function's scope. If your file only defines classes, functions, constants etc. that are scope-independent, this will work fine. (Refer to my original answer for importing variables from files using a helper function.)
Edit: It turns out OP wasn't asking about including multiple files. My original answer has been ported over to: How to include() all PHP files from a directory?.