Quorum Drive needed?
No, you do not need a quorum disk for availability groups or failover instances.  Microsoft Failover Clustering has supported the file share witness since Windows 2003 SP1.  The file share witness, in conjunction with majority node set, enables a cluster quorum to operate with no shared disk infrastructure.

Technically, the cluster can function with majority node set only, but the typical availability group configuration also uses at least one file share witness so that a node gain take ownership of quorum when one or more other nodes lose communication.

With a failover instance, a shared disk is required for the SQL Server program and data files.  A quorum disk would typically be configured for a failover instance because the shared disk infrastructure is already required.  However, it would not be required and the a majority node set / file share witness quorum could be used.

See Configure and manage quorum and  Clustering: What exactly is a File Share Witness and when should I use one? for more details.