While FIOS does not use modems, a MOCA WAN bridge can serve a similar purpose.
»Verizon Online FiOS FAQ »What is a MOCA Bridge?

Switching the ONT from coax WAN to cat5 WAN is the preferred approach for connecting a third party router.
»Verizon Online FiOS FAQ »Replacing the Actiontec (part 1): Coax to Ethernet
However, switching to a cat5 WAN connection is not always practical. When keeping the coax WAN connection is the only option, a MOCA WAN bridge can provide a solution. By placing a MOCA WAN bridge between the ONT and a user router, one can create a layer two bridge between the coax WAN and the cat5 WAN port of a user router.

Two MOCA bridges that MOCA WAN are available from various sources (eBay, Newegg, Amazon, etc.
• Netgear MCAB1001 * - available only in pairs.
• D-Link DXN-221 * - Pair or DXN-220 single.
Note that the Motorola NIM-100 and Actiontec ECB2200 are not suitable as they do not support the MOCA WAN channel.

The following instructions were tested with the Netgear MCAB1001. Instructions for the D-Link DXN-221 should be similar.
If someone goes through this procedure with the D-Link unit, please post exact steps.

Prepare the Netgear MCAB1001
Do not connect the MCAB1001 to the coax yet.
• Connect a PC to the RJ-45 jack on the bridge.
• Place the bridge in CONFIG mode.
• Run the supplied disk to configure the bridge.
• On the COAX page, select channel 9 (1000 Mhz)
• Set the Diplexer Mode to ALL PASS
• Press APPLY
• Disconnect the PC from the bridge
• Release the CONFIG button

Configuring the Actiontec
These steps are necessary if you have FIOS-TV, since the MOCA LAN connection from the Actiontec is still required for the STBs.
If you do not have FIOS-TV, you can simply disconnect the coax from the Actiontec.
• Login to the Actiontec router.
• Click on the My Network icon.
• Select Network Connections from the left menu.
• Select Broadband Connection (coax) from the connection list.
• Click on SETTINGS
• Across from DHCP Lease, click RELEASE
• Click CANCEL
• Click on DISABLE

Connecting the bridge
• Connect the coax side of the MOCA bridge to the main splitter.
• Connect the ethernet side of the MOCA bridge to the WAN port of your router.
Your router should now pull a public DHCP lease from Verizon's DHCP server.

D-Link DXN-220
For those using the DXN-220, details instructions can be found here:
»Re: Testing MOCA bridges -- update