Broken Nexus 6p Recovery
Yes, you can. Connect your phone to a PC and use ADB to backup your data. Follow this guide to setup ADB, Fastboot and the Android SDK Platform tools on your PC: https://forum.xda-developers.com/nexus-6p/general/guides-how-to-guides-beginners-t3206928

Load your phone into fastboot mode by holding the power button and the volume down button at the same time, from there you can choose recovery mode by the volume and power buttons navigation of your phone, then access and copy your data from your PC by ADB. 

Alternatively you could temporarily load a custom recovery as suggested here:  can fastboot be used to backup device?

For more Info see this answer here: Enabling ADB on Nexus 6 with dead screen