You can use the cluster node initiaization script to grab the environment variable DB_CLUSTER_ID. Refer here https://docs.databricks.com/user-guide/clusters/init-scripts.html#environment-variables. Also from the notebook, the following also works :

spark.conf.get("spark.databricks.clusterUsageTags.clusterName")

spark.conf.get("spark.databricks.clusterUsageTags.clusterId")