It took me awhile to figure this out so I figured I would share so others don't have to redo this. I wanted my contacts updated on the latest 22.214.171.124 desktop version of Verizon Messages.
1) Export google contacts in a vCard (for iOS Contacts). For me export was on the left side under the "More," then click on export.
2) Import this file into your Verizon Cloud. It's free for just contacts. You might want to delete them all first if all you want is the latest contacts you have in google contacts. I was able to import 902 of my 942 contacts. I'm not sure why 40 were left off though I know a few had more than a 1st and last name in those fields. I manually updated a few myself on the Verizon cloud.
3) Install/open the 126.96.36.199 version (or maybe any 1.x version?). You can find copies out on the internet. I happened to still have the install file. The after it updates, click the three bars on the top right and select sync.
4) Then open the 188.8.131.52 version and it should update your contacts. You can uninstall the 1.x version. I left mine on for now though the popup windows when clicked now open up the 1.x version.