To get the Add method, you need to first click the Explore Component Properties button in the upper right of the Object Explorer window. From here you can locate the Items property and select its Add method.
What you want to do in this case though is to add a TableView to your Windows Form. Then you can set its TableProvider property to that of your SQL Query component. You can then set the DataSource property of your ListBox to the TableView.
To move items between the list, you'll need to Add() the SelectedItems to the second listbox. Basically, pass the SelectedItems property from the first listbox into a ForLoop and call the Add() method on the second listbox for each of the items (or you may want to use the Insert() method to preserve order).