We two are going to Vancouver with a transfer in Narita Airport, Japan, and need to exit from Terminal T3 and enter at Terminal T1 to handle the transfer.

No, because you are transiting within Japan, you only need to complete the transfer procedures from T3 Terminal to T1 Terminal at Narita Airport, and you do not need to apply for a Japanese visa. But please make sure you have a valid international travel document in your passport as well as a transit visa (if required by the destination country).