VZ does not use modems except in certain MDU VDSL installations. If you have a VDSL modem, no you can not install a second VDSL modem.
If you're referring to the Actiontec MI424-WR ROUTER (it is not a modem), then yes, you can install a second router. See the instructions here:
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Verizon FiOS FAQ »
Can I get an ethernet connection in a room with only coax?No, it won't affect your bill.