I was able to find what I needed by enabling UCS Manager XML logging (Ctrl-Alt-Q twice), navigating to the vNIC template, and then the VLAN Groups to view the VLAN groups assigned to the vNIC template. This logged the information I needed to find a solution.
Here is what the xml log reported
<configResolveChildren
cookie="164xxxxxx"
classId="fabricNetGroupRef"
inDn="org-root/ls-ESXserver/ether-CORP-Compute-B"
inHierarchical="false">
<inFilter>
</inFilter>
</configResolveChildren>
The classId provided the clue to the cmdlet to use: "fabricNetGroupRef"
If you do a get-command -name *FabricNetGroupRef* -Module Cisco.UCSManager you will see the cmdlet
Get-UCSFabricNetGroupRef
Starting from a Service Profile Template you can drill down to the VLAN Group(s) assigned to the vNIC templates associated with that template as follows:
Get-UcsServiceProfile|Get-UCSVNIC|Get-UCSFabricNetGroupRef