Its been so late that Nexus 6 by Verizon getting Android 5.1.1 update now, as all Nexus 6 already gets Lollpop 5.1.1 update. As the carrier based Nexus always be late in providing update compared to non carrier based.
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You can Update your Google Nexus 6 by going to Settings> About Phone and Check for Update. The Latest software version is LMY47Z.
Its better if you try with WiFi network and then download the update.
So enjoy Android 5.1.1 Lollipop update and also tell us how you feel after updating it.