It's strange to me to embed conditions in attributes - particularly the v-else case in a different tag. It requires several visual scans of the block of code to determine what's going on.

In my opinion the following requires less cognitive overhead:

    function Home(props) {
      var query = props.location.query.home ?
        ( <p>You specified a home query: { props.location.query.home }</p> )
        : ( <p>No home query was specified.</p> );
      return (
        <div>
          <h2>HELLO</h2>
          <p>Duis a turpis sed lacus dapibus elementum sed eu lectus.</p>
          {query}
        </div>
      );
    }