It's quite clear when the Cube is first hit, and when you see the Defiant firing its phaser cannons, that every shot is breaching their hull. The Enterprise takes out their tractor beam with 2 shots. Where are their shields?
The Borg do have shields, just not ones that envelop the entire vessel, preferring to shield their most critical systems independently. Their exterior hull is heavily armoured, though and the majority of damage to the vessel was largely cosmetic and would have been easily repaired had Picard not known where to strike them:
The whisper of the one and the multitude grew briefly louder.
critical damage to shields at power sector one-one one. All drones coordinate repair immediately....
The mental whisper died abruptly, as if the speakers had realized he was listening. But it was too late; he had already experienced a revelation beyond the mere words he had detected.
The ship did have shields. However the cube sustained substantial damage on it's trip to Earth battling Starfleet; as such it's shields were weakened to the point it sustained hull damage, however borg ships are not homogenized so hull damage is largely inconsequential. If you listen to the film Picard: What is the status of the Borg Cube? DATA: It has sustained heavy damage to its outer hull. I am reading fluctuations in their power grid.