If you have the Verizon Cloud feature on your mobile number, open the Cloud app, go to settings (tap the menu icon on the top left first), tap What to backup and tap Messages. You should know the first backup will only go back 90 days and anything longer won't be backed up.
Or, if you are using the Verizon Message+ app, I think they might backup more. Im only familiar with the Cloud app.
We want to ensure your messages are stored securely, D03sb1. In order to better assist, I have a few questions. What is the make and model of device that you use? What messaging application are you currently using? Looking forward to hearing back.