If you are experiencing this, something likely went wrong with your installation.  Please follow the steps below EXACTLY. 

1) The main quest in FCR, which you begin by talking with Peter, will not start unless you have completed at least ONE of the following: the Railroad quest "Tradecraft" (with Deacon), the BoS quest "Tactical Thinking" (destroy the Railroad), The Institute quest "End of the Line" (destroy the Railroad), the Minutemen quest "Inside Job" (Institute holotape to Sturges), or the main quest "Hunter/Hunted".

2) Console command: setstage 6FA37 900.  This will set the quest Tradecraft as complete, meeting the conditions for starting the quest Remnants of Vault 59 that Peter gives you.  If your only issue is that the quest "Remnants of Vault 59" isn't showing, the console commands to force start it are here (e.g. setstage xx000F9C 5, and replace the xx).

3) If basic stuff like voice files aren't working, or Peter has no face (only teeth and eyes), you've got an installation issue.  The same thing happens to a small percentage of users who try to install large mods like CBBE using Nexus Mod Manager, as the most recent versions of NMM sometimes have trouble installing large files.

To avoid problems with missing audio/texture/meshes when installing with NMM, do NOT install using the green buttons on the main mod page (e.g. Download NMM button).  Navigate to the files page, and choose either "Download with Manager" or "Download Manually".  "Download Manually", and then dragging the file into your mod manager, is the most trouble-free way of doing it.

It is highly recommended that you run NMM in admin mode.

If you use the multiple hard drive install option with NMM, e.g. your game is installed on your SSD drive while your mods are stored on a separate HDD drive, this can create problems with the game accessing various asset files, in versions of NMM later than v61.23.  Manually placing the actual files in the data folder should fix it.

Here are full troubleshooting steps:

A) In meshes/actors/character/characterassets, make sure the following file is showing up:

ChildMaleHeadPeter.nif

B) In textures/actors/character/HumanChildren, make sure that the following files from Fusion City Rising are showing up (from Fusion City Rising, not another mod):

ChildMaleHead_s.dds
ChildMaleHeadPeter.dds

If those files are missing, or are present but the game isn't recognizing them, manually drag those files, and any voice/music files from Fusion City Rising, from the Fusion City Rising file into your game's data folder. 

If you aren't comfortable with that approach, here is the reinstall procedure with NMM:

In NMM, completely uninstall Fusion City Rising (not just deactivate, completely uninstall so it is removed). Download FCR from the files page using "Download Manually", and drag it into your mod manager.  Do NOT use the "Download NMM" green button on the main mod page.

Then re-install using NMM.

Fallout 4 Nexus Virtual install sometimes has a problem with custom audio from mods, which can cause Peter to have no voice, and is why you should always download via the files page.  AGAIN, DOWNLOAD VIA THE FILES PAGE, NOT THE GREEN BUTTONS AT THE TOP OF THE MOD PAGE.

When installing, let Fusion City Rising overwrite textures from any other child mods, or try "yes to all".

After re-install, load a save from BEFORE you had Fusion City Rising installed.

If still having issues:

Re-read and follow the above instructions EXACTLY.  If you shortcut it, there is not much we can do to help.

Again, running NMM in admin mode is another alternative that has worked for a couple people who had an issue with NMM not installing files properly.

You can also try reverting to an earlier, less buggy version of Nexus Mod Manager.  Version 61.23 of NMM was arguably the last release that seemed to be free of most installation headaches:
http://nmm.nexusmods.com/Releases/4.5/Nexus%20Mod%20Manager-0.61.23.exe

Also see the load order notes here.  If nothing works, and you're using a child mod other than the ones on the list from here, try putting that mod below Fusion City Rising.

If still no luck, if you open up all your mods in FO4 Edit, are any of them altering the Race "HumanChildRace" (FormID: 0011D83F)?