machbook pro 2013 a1425 connected to internet but no software update

What software update are you expecting? a1425 corresponds to a MacBook Pro10,2 13" late-2012/mid-2013 model - which cannot be upgraded beyond macOS Catalina.

If you mean you can't access websites using Safari, that is pretty much expected as Safari is also not updated. You should switch to another browser that is getting updates - Firefox, for example. There are lots of choices.

Are you running any VPN, Anti-Virus, or Cleaning apps?

Make a New Location, Using network locations in Mac OS X ...

System Preferences>Network, top of window>Locations>Edit Locations, little plus icon, give it a name.

System Preferences>Network, click on the little gear at the bottom next to the + & - icons, (unlock lock first if locked), choose Set Service Order.

The interface that connects to the Internet should be dragged to the top of the list.

Sys Prefs>Network>highlight the Interface you're using>Advanced>TCP/IP>Configure IPv4: Using DHCP.

If using Wifi, instead of joining your Network from the list, click the WiFi icon at the top, and click join other network. Fill in everything as needed.

System Preferences>Network>choose interface>Advanced>Proxies Tab, make sure none are set, like for HTTP & HTTPS.

System Preferences>Network, unlock the lock if need be, highlight the Interface you use to connect to Internet, click on the advanced button, click on the DNS tab, click on the little plus icon, then add these numbers...

8.8.8.8

8.8.4.4

1.1.1.1

9.9.9.9

Apply.