Work, yes. LTE, no.
If you're already on 4.1.2, I strongly suggest you make the same backups as if you were to flash the LTE modems if you're going to tinker with the original 4.1.1 radio.
If you really don't want to bother with rooting and flashing for now and can wait a bit, my best guess is that Videotron will have an LTE update out by the time they flip the switch on their LTE deployments, supposedly by the end of the year.
My phones: Galaxy Note II SGH-T889V (with LTE enabled
/ ext4 SD card support on TouchWiz ROMs
Previous phones: Nexus S GT-I9020A, Galaxy Gio GT-S5660M (Arpeggiomod ROM
/ Discussion, stock rooted ROM
Phones for which I play "tech support" for family and friends: Galaxy Nexus GT-I9250, Galaxy Q SGH-T589R, Galaxy 551 GT-I5510M, LG C-800G Eclypse
- Level 8