Backup/restauração do banco de dados SQLlite para a pasta do aplicativo Google Drive

 ... 
 saveToDrive(
   Drive.DriveApi.getAppFolder(getGoogleApiClient()),
   "MyDbFile.db", 
   "application/x-sqlite3",
   new java.io.File("\...\...\...\MyDbFile.db")
 );
 ...

DriveId mDriveId;
/******************************************************************
 * create file in GOODrive
 * @param pFldr parent's ID
 * @param titl  file name
 * @param mime  file mime type  (application/x-sqlite3)
 * @param file  file (with content) to create
 */
void saveToDrive(final DriveFolder pFldr, final String titl,
                 final String mime, final java.io.File file) {
  if (getGoogleApiClient() != null && pFldr != null && titl != null && mime != null && file != null) try {
    // create content from file
    Drive.DriveApi.newDriveContents(getGoogleApiClient()).setResultCallback(new ResultCallback<DriveContentsResult>() {
      @Override
      public void onResult(DriveContentsResult driveContentsResult) {
        DriveContents cont = driveContentsResult != null && driveContentsResult.getStatus().isSuccess() ?
          driveContentsResult.getDriveContents() : null;

        // write file to content, chunk by chunk
        if (cont != null) try {
          OutputStream oos = cont.getOutputStream();
          if (oos != null) try {
            InputStream is = new FileInputStream(file);
            byte[] buf = new byte[4096];
            int c;
            while ((c = is.read(buf, 0, buf.length)) > 0) {
              oos.write(buf, 0, c);
              oos.flush();
            }
          }
          finally { oos.close();}

          // content's COOL, create metadata
          MetadataChangeSet meta = new Builder().setTitle(titl).setMimeType(mime).build();

          // now create file on GooDrive
          pFldr.createFile(getGoogleApiClient(), meta, cont).setResultCallback(new ResultCallback<DriveFileResult>() {
            @Override
            public void onResult(DriveFileResult driveFileResult) {
              if (driveFileResult != null && driveFileResult.getStatus().isSuccess()) {
                DriveFile dFil = driveFileResult != null && driveFileResult.getStatus().isSuccess() ?
                  driveFileResult.getDriveFile() : null;
                if (dFil != null) {
                  // BINGO , file uploaded
                  dFil.getMetadata(getGoogleApiClient()).setResultCallback(new ResultCallback<MetadataResult>() {
                    @Override
                    public void onResult(MetadataResult metadataResult) {
                      if (metadataResult != null && metadataResult.getStatus().isSuccess()) {
                        DriveId mDriveId = metadataResult.getMetadata().getDriveId();
                      }
                    }
                  });
                }
              } else { /* report error */     }
            }
          });
        } catch (Exception e) { e.printStackTrace(); }
      }
    });
  } catch (Exception e) { e.printStackTrace(); }
}

/*******************************************************************
 * get file contents
 */
void readFromGooDrive() {
  byte[] buf = null;
  if (getGoogleApiClient() != null && getGoogleApiClient().isConnected()) try {
    DriveFile df = Drive.DriveApi.getFile(getGoogleApiClient(), mDriveId);
    df.open(getGoogleApiClient(), DriveFile.MODE_READ_ONLY, null)
      .setResultCallback(new ResultCallback<DriveContentsResult>() {
      @Override
      public void onResult(DriveContentsResult driveContentsResult) {
        if ((driveContentsResult != null) && driveContentsResult.getStatus().isSuccess()) {
          DriveContents cont = driveContentsResult.getDriveContents();
          // DUMP cont.getInputStream() to your DB file
          cont.discard(getGoogleApiClient());    // or cont.commit();  they are equiv if READONLY
        }
      }
    });
  } catch (Exception e) { e.printStackTrace(); }
}