Abaixo, o código está funcionando bem em dispositivos pré-Marshmallow, mas não no Marshmallow.

Essas são as permissões no manifesto

<uses-permission android:name="android.permission.INTERNET" />
<uses-permission android:name="android.permission.READ_EXTERNAL_STORAGE"/>
<uses-permission android:name="android.permission.WRITE_EXTERNAL_STORAGE" />

Aqui está o código

public void saveImageToSDCard(Bitmap bitmap) {
    File myDir = new File(
            Environment
                    .getExternalStoragePublicDirectory(Environment.DIRECTORY_PICTURES),
            pref.getGalleryName());

    myDir.mkdirs();
    Random generator = new Random();
    int n = 10000;
    n = generator.nextInt(n);
    String fname = "Wallpaper-" + n + ".jpg";
    File file = new File(myDir, fname);
    if (file.exists())
        file.delete();
    try {
        FileOutputStream out = new FileOutputStream(file);
        bitmap.compress(Bitmap.CompressFormat.JPEG, 90, out);
        out.flush();
        out.close();
        Uri uri = getImageContentUri(_context,file);

        Log.d(TAG, "Wallpaper saved to: " + file.getAbsolutePath());

    } catch (Exception e) {
        e.printStackTrace();
    }
}

E o mesmo código funciona quando eu permito manualmente a permissão de armazenamento

Aqui está a solução dada por Nitesh Pareek.

private boolean hasPermissions(Context context, String[] permissions) {
    if (android.os.Build.VERSION.SDK_INT >= Build.VERSION_CODES.M && context != null && permissions != null) {
        for (String permission : permissions) {
            if (ActivityCompat.checkSelfPermission(context, permission) != PackageManager.PERMISSION_GRANTED) {
                return false;
            }
        }
    }
    return true;
}
String[] PERMISSIONS = new String[]{ Manifest.permission.WRITE_EXTERNAL_STORAGE};

    if (!hasPermissions(this, PERMISSIONS)) {
        ActivityCompat.requestPermissions(this, PERMISSIONS, 11);
        return;
    }