I'm glad to help you transfer your data from QuickBooks Desktop (QBDT) to QuickBooks Online (QBO).
Yes, you can export out your QBDT 2013 data to QBO. Since your QBDT version is outdated, you may need to use an upgraded/latest version of the software. But no worries, you can download a free trial version and use it in transferring your files.
Before that, secure a backup copy of your QBDT company to save the data. I'll guide you on how.
Go to the File menu, then choose Back Up Company.
Click on Create Local Backup.
Select Local Backup. Then, click Options.
In the Tell us where to save your backup copies field, select Browse and select the location in which to save the backup copy.
In the Create Backup window, specify whether to back up automatically, and select Options to set options for automatic backups.
Tap New to set a backup schedule in the Schedule Backups dialog.