Transferring Apple Music from Verizon to Apple One

If both of your subscriptions are linked to your account then when your Verizon subscription expires it appears that it just continues on Apple One. That's my reading of:

Connect Apple subscriptions from your wireless carrier or other provider to your Apple Account - Apple Support

If you subscribe to an Apple service billed through Apple and then receive the same subscription from your wireless carrier or other provider, cancel one to avoid duplicate charges. You can cancel your subscription from Apple or cancel your subscription through your wireless carrier or other provider.

The only way I can think of that might work is to start Apple One before the Verizon subscription ends. Even then I don't know if it will be regarded as the same subscription only with separate payment methods, or if it will be seen as ending one and starting the other.