Uma das formas de se gerar backup do banco de dados do seu aplicativo é utilizando os comandos abaixo.

public void gerarBackup(Boolean silent) {
	try {
		File sd = Environment.getExternalStorageDirectory();
		File data = Environment.getDataDirectory();

		if (sd.canWrite()) {

			createDirIfNotExists("/bkps");

			String currentDBPath = "//data//br.com.j2v.app.nomeAplicativo//databases//dbNAME";
			String backupDBPath = "bkps/dbNAME-" + getDateTime("log") + ".db";
			File currentDB = new File(data, currentDBPath);
			File backupDB = new File(sd, backupDBPath);

			FileChannel src = new FileInputStream(currentDB).getChannel();
			FileChannel dst = new FileOutputStream(backupDB).getChannel();
			dst.transferFrom(src, 0, src.size());
			src.close();
			dst.close();
			if (!silent) {
				Toast.makeText(getBaseContext(), "Backup gerado com sucesso!\r\nCaminho: " + backupDBPath, Toast.LENGTH_LONG).show();
			}
		}
	} catch (Exception e) {
		Log.e("J2V", e.toString());
		Toast.makeText(getBaseContext(), e.toString(), Toast.LENGTH_LONG).show();
	}
}