Yes, you can go crabbing in Newquay. In fact, it's a popular pastime with visitors and locals alike along the walls of Newquay Harbour, although it's also possible in other areas too - just be careful and avoid climbing over any slippery rocks or cliffs.
Sustainable tourism is important to Cornwall locals, so we'd always encourage releasing your catch when you're finished and being aware of the regulations protecting smaller juvenile crabs. Make sure your line is strong and secure too, so you're not at risk of losing anything in the sea, particularly plastic.