Can you confirm that the Moodle site's domain name resolves to the same IP address at the office and at home?
If you telnet to the IP address on port 443 (or port 80 if it's not an HTTPS site) from the office does this fail to connect? What message do you get?
Traceroute is an inexact test as some hops may deliberately not respond. But I can't see that the office connection converges on the route taken by the home connection which I'd expect.
There are two potential issues with routing I can think of: 1) the routing tables used on the outbound route and 2) the routing tables used on the return route.
If it is a routing issue then first check with the local network support that there are no specific routing rules for the Moodle site's IP address.
If you manage or have root access to the Moodle web server check that the routing table on this has no specific rule which may cause this. If possible run a traceroute from the server back to the public IP addresses of your home and office and compare these.
The web server (e.g. Apache, Nginx) could have rules blocking specific IP addresses.
Any routing issue that was external to both your office and the Moodle web server would have to be escalated to the ISP for your office's Internet connection or that hosting the web site.
I'm assuming there's no web proxy in use at your office, if there is you'll need to be doing your nslookup, telnet and routing tests to/from this.