Verizon dragged their feet offering the Nexus 6 officially, but it was still possible to use the phone on their network if you picked up the unlocked Play Store model and popped in a Verizon SIM card. One issue with this according to some users was that using both voice and data at the same time when connected to Verizon's network was not a possibility, which may have posed a bigger problem to some users than they realized. It seems that the recent update to Android 5.1 on the Nexus 6 has done away with this issue for Verizon customers, and use of data while on a voice call is now a capability at anyone's disposal on the Verizon network who has a Nexus 6 running on Android 5.1.
It isn't 100% straightforward, and things don't necessarily just work, as Droid Life points out that there may be a couple of things users will have to do to get things up and running. In fact, there is a small list of steps to carry out if you want simultaneous voice and data turned on. To begin, users will need to enable two voice related modes called Enhanced 4G LTE Mode as well as VoLTE. The first is technically located in the phone settings menu but it isn't immediately within view, so to grab access to the menu where you can turn on the enhanced 4G LTE function you have to first hit the following sequence of buttons on the dialer, *#*#4636#*#*, after which you'll be taken to the phone settings menu that contains the option to turn on VoLTE listed as turn on VoLTE provision flag.
Follow this up with a phone reboot and then head directly to the phone settings menu from within settings, tap on wireless and networks, hit the more button to bring up additional options and make sure the newly visible option of enhanced 4G LTE mode is turned on. Getting this to work may require users to add the free HD voice feature onto their plan, which can be done from within their Verizon account. That should be all there is to it so long as all the steps were done and your device is running 5.1 Lollipop.