I have a computer(1) which is connected via TeamViewer to another computer(2), in another network. My question is, is it possible to somehow monitor (1)'s network activity on (2)?
It's not possible literally - without somehow forcing all traffic for computer(1) through computer(2) - but doing this using TeamViewer would be very inefficient.
You can of course install network monitoring tool on computer(1) and view it's output through the TeamViewer connection...
The only activity related to computer(1) you can monitor on computer(2) are packets directed from computer(1) to computer(2). The TeamViewer software creates VPN (Virtual Private Network) to which both of the computers are connected. However other traffic is never routed through this virtual network - as there is no reason for it.