They're altogether different. GPU computing is a variation of standard electronic computing which uses GPU cores to execute calculations, generally for programs optimized for highly parallel computation. GPUs feature very large numbers of relatively weak cores compared to CPUs but for certain types of operations friendly to parallel computation work goes much faster on a GPU. Not all programs can be accelerated this way, since some are by nature serial (instructions following in a strict order).
Quantum computing is a different beast. It doesn't use discrete bits as we know them. 'Qubits' (quantum bits) exist in a "superposition" of both 0 and 1 until their state is resolved. One of the tricks is to maintain the qubits in this quantum state, as they need to be very isolated from outside influences. The state of the art in quantum computations as now is no further than basic arithmetic with small numbers. With larger numbers of qubits they get very powerful though, and have profound implications for cryptography among other things. A quantum computer may be able to easily break any of the cryptographic keys used to encrypt data and online communications.