You use a packet sniffer (wireshark, tcpdump) to capture traffic, and the OS tools (netstat) to determine the network port used by the Chrome process (each Chrome tab is its own process). Then you simply pair the traffic from this process to its destinations.
But the premise is ridiculous. If Chrome wanted to secretly copy the data to its own servers so that ...
You may also use pyshark module for Python. You will find certificates in the SSL layer, but not all SSL layer packets have the certificate. Some of the packets are for handshake negotiation, etc. So, you can search for one of the x509 attributes in the packet.
# Import modules
# Docs: https://github.com/KimiNewt/pyshark/