I assume you have installed Debian in a way that it uses it's own separate swap partition. If you have not, I would suggest doing so (create another swap partition and change the
/etc/fstabfiles accordingly), because too different operating systems writing to the same swap partition may discard hibernation data stored on the partition by the other OS.
Then you have to ensure that the OS you want to use hibernation on (usually both), writes to the correct partition. This is configured in the
/etc/initramfs-tools/conf.d/resumefile. You have to put the UUID of the correct swap partition in there (use
sudo blkid /dev/$device_nameto get the UUID). Finally you need to update the initial ramdisk:
sudo update-initramfs -u -k all