In Java how can I add an object to an array via the object's constructor?

You are asking more about general concepts of programming, not Java itself.

In short - constructor's sole responsibility is to create and initialize an object. Doing anything else in a constructor would be a bad practice.

You can use a factory pattern though:

public class BookCollection {
    public static final int MAX_COUNT = 20;

    private int count = 0;
    private Book[] books = new Book[MAX_COUNT];

    public Book createBook(String nameP, String authorP, String genreP, int ratingP, int iDP, Boolean onLoanP) {
        books[count] = new Book(nameP, authorP, genreP, ratingP, iDP, onLoanP);

        return books[count++];
    }

    public Book[] getBooks() {
         return books;
    }
}

This implementation is open for many problems - like too many books created, concurrent access issues and many others.

But should give you a guidance, where to go.

Personally I would rather go with a collection, such as ArrayList. Not an array. Arrays are great in some cases, but most of the time an ArrayList will be much better.

public class BookCollection {
    private List<Book> books = new ArrayList<Book>();

    public Book createBook(String nameP, String authorP, String genreP, int ratingP, int iDP, Boolean onLoanP) {
        Book result = new Book(nameP, authorP, genreP, ratingP, iDP, onLoanP)

        books.add(result);

        return result;
    }

    public List<Books> getBooks() {
         return Collections.unmodifiableList(books);
    }
}

If you use primitive arrays (i.e., Books[]) instead of data structures like List/Vector, I think you can use normal array init. e.g.,:

Book[] books = {new Book(bla,bla,bla), new Book(foo,foo,foo),... };

However, I tend to prefer data structures, and there are ways to initialize those with constants, depending on the structure and the library.

Caveat: I recognize that the entire approach here is problematic, I'm just answering the question as asked.

Stick everything in an arraylist, let it manage its own size and then dump the contents into an array sized off the arraylist.

public Book(String title, String author, String genre, int rating, 
            int id, boolean onLoad, ArrayList<Book> array)
{
    //initialize stuff
    array.add(this);
}

In the main code where you're setting up the objects:

ArrayList<Book> tmpArray = new ArrayList<>();
//build your Book objects, passing in tmpArray
Book array[] = tmpArray.toArray(new Book[tmpArray.size()]);

And you have the array that you desired.

// Poor
private Book[] books = new Book[5];
...
Book b1 = new Book("Catch 22", "Heller", "Thriller", 5, 1, false);
Book b2 = new Book("The Hobbit", "Tolkien", "Fantasy", 4, 2, false);
...
books[0] = b1;
books[1] = b2;

// Better
Book[] books = new Book[] {
  new Book("Catch 22", "Heller", "Thriller", 5, 1, false),
  new Book("The Hobbit", "Tolkien", "Fantasy", 4, 2, false)
};
...

// Better still
private ArrayList<Book> books = new ArrayList<Book>();
books.add(new Book("Catch 22", "Heller", "Thriller", 5, 1, false);
...