Not unless you have a smartphone and have them set to back up automatically or send to another phone automatically. If you have a basic phone, the odds are even slimmer. If you have a smartphone, you can search the respective app market for a SMS backup/forwarding tool which may provide the service you would like to have.
Verizon only keeps a week or so of texts on their servers at any one time, and you cannot get the details from those without a court order. You can sign into your MyVerizon account if you are the account owner and see information for the last 3 months which would include time sent/received and the number the text was to/from.
You can't see the text messages, but you can see the text logs ... current billing cycle logs can be found in your on-line MyVerizon account under Usage Details (in blue). Previous billing cycles can be found under View Bill, then use the drop-down to select the timeframe. Then use the appropriate tabs. Text/data logs are available for the past 90 days; bills are available for the past 12 months.