Need to update safari browser on my mac 10.13.6

If you can't upgrade your MacBook Pro, then for security reasons I would recommend that you use a 3rd party browser that supports running their current version on macOS 10.13.6. One such option is Chrome.

Also:

The model year of a MacBook Pro is available at Apple menu > About This Mac.

A mid 2010 through late 2011 MacBook Pro can be upgraded to High Sierra 10.13.6 but no later version. See: How to download macOS - Apple Support

A mid 2012 through early 2013 MacBook Pro can be upgraded to Catalina 10.15.7 but no later. See: How to download macOS - Apple Support

A late 2013 and mid 2014 can be upgraded to the latest  version of Big Sur macOS 11 but no later. See:How to download macOS - Apple Support

An early 2015 through 2016 MacBook Pro can be upgraded to the latest version of Monterey macOS 12. See: How to download macOS - Apple Support

A 2017 or later MacBook Pro can be upgraded to the current version of Ventura macOS 13. See: How to upgrade to macOS Ventura - Apple Support

Be aware that any upgrade to Catalina or later will also require updating any non-Apple 32-bit apps (if you have any). See: https://www.macworld.com/article/232735/how-to-check-if-mac-software-is-32-or-64-bit.html