Depending on what mechanism is used to send emails and it's configuration, you might be able to find some logs of sent emails. However, if your machine has been compromised and is sending using some sort of malware, logs are unlikely.
The only sure-fire way to find out is to monitor the network traffic flowing in and out of the machine. You can run a network sniffing tool like wireshark to do this, although it is not a simple tool to use.
To be totally sure, you need to run the network monitoring software between this machine and the rest of the network - if the machine is compromised you can't trust anything it tells you, including what network traffic is being sent/received.