Is this the only or best method for classes serializing in Java?
All of your variables should be local, except for maybe the FileChooser.

You swallow exceptions.  Printing a stack trace is no substitution for proper exception handling.  (By the way, logging should be done using java.util.logging.  If you must print a stack trace manually, print it to System.err instead of System.out.)  Serialization could completely fail, and the function will still return true.  The most appropriate behaviour should be to propagate the exception to the caller.

For greater flexibility and clarity with little additional effort, I suggest overloading serialize() and deserialize() with several variants which call each other:

public class InputOutput {
    private static final FileChooser seleccionador = new FileChooser();

    /**
     * Serializes an object to a file chosen by the user using a FileChooser.
     * @return true if completed, false if user cancels the FileChooser.
     */
    public static boolean serialize(Object o) throws IOException {
        seleccionador.setTitle("Dónde guardar el estado del programa");
        File f = seleccionador.showSaveDialog(null);
        if (f == null) return false;
        serialize(o, f);
        return true;
    }

    public static void serialize(Object o, File f) throws IOException {
        try (FileOutputStream fos = new FileOutputStream(f)) {
            serialize(o, fos);
        }
    }

    public static void serialize(Object o, FileOutputStream fos) throws IOException {
        try (ObjectOutputStream oos = new ObjectOutputStream(fos)) {
            oos.writeObject(o);
        }
    }

    /**
     * Deserializes an object from a file chosen by the user using a FileChooser.
     * @return the deserialized object, or null if user cancels the FileChooser.
     */
    public static Object deserialize() throws IOException { ... }
    public static Object deserialize(File f) throws IOException { ... }
    public static Object deserialize(FileInputStream fis) throws IOException { ... }
}