First, if you have a red dot on the Trezor logo (bottom right) - yes, Electrum has a communication problem with your Trezor - check the cable. Sometimes, it could be the laptop or PC USB connector that can be worn out or "loose" (my 7 years old laptop has that issue).
Second, even if you had the green light on the Trezor logo, you can't export private keys out of Trezor itself.
The only way you can access the private keys stored on a Trezor, is to restore the seed words into a regular Electrum wallet. But, by doing so, you will compromised your seed words - therefore, not recommended.
Thank you. The reason that I'm asking is because I have some non-bitcoin tokens stored on it that are compatible with bitcoin addresses and private keys, so I need the private key in order to import the tokens into their native wallet.
So in your view using the wallet seed when electrum starts would do the trick and generate those private keys?