Yes, the LG G4 uses a micro SIM and the Note 5 uses a nano. If the Note 5 didn't have a SIM preinstalled when you purchased it, one option is to purchase a SIM card adapter so that you can use the nano SIM in any phone. Another option is to get the correct size SIM card; they're free at your local Corporate VzW store or by contacting Customer Service.
I regret the delay as I understand how important it is to stay connected on the main phone line. The Note 5 does take a different SIM. CAn log in via http://www.vzw.com/myverizon and update the ICCID(SIM card number) on the needed line to enable the Note 5 ICCID. Does the G4 show any physical or liquid damage? Have you tested multiple chargers including the original?
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