Can porcupines shoot their quills?

No. Porcupines can't shoot their quills. The quills are hollow, modified hairs that are only loosely attached, so they pull free on contact and stay lodged in a predator, which makes it look as if they were fired.

What's actually true

Porcupines can't shoot their quills. No porcupine can fire quills like darts. The quills are hollow, modified hairs that are only loosely attached to the skin. When a predator bites or brushes against a porcupine, the quills pull free easily and stay lodged in the attacker, which makes it look as if they were launched.

Takeaway: Porcupines defend by contact, so keeping distance keeps you quill-free.

Why people believe it

Usually nobody sees the moment of contact. A dog comes back with a face full of quills, or an animal is found with quills stuck in it far from the porcupine, and 'it shot them' is the simplest story. Porcupines also swing their tails and shake their quills as a warning, which can look like an attack. In fact the porcupine does not need to aim: the barbed tips and loose attachment do all the work.

Source