Inspect firmware update using RACADM command-line utility which provides a scriptable interface for Remote Access Controller.

Inspect job queue for firmware update tasks.

$ sudo racadm jobqueue view | \
  awk -v FS="\n" -v RS="----------------------------------------------------------" '$0~/Firmware Update/ {print $0}'
[Job ID=JID_231925652951]
Job Name=Firmware Update: SAS RAID
Status=Scheduled
Start Time=[Next Reboot]
Expiration Time=[Not Applicable]
Message=[JCP001: Task successfully scheduled.]
Percent Complete=[0]

Simplify it a bit to detect if reboot is required.

$ sudo racadm jobqueue view | \
  awk -v FS="\n" -v RS="----------------------------------------------------------" '$0~/Firmware Update/ {print $5}'
Start Time=[Next Reboot]