In classical cryptography, security proofs are often based on the (assumed) computational hardness of some mathematical problem. Using the principles of quantum mechanics might provide means to design cryptographic protocols for which it is impossible to realise them classically (information-theoretically) securely. But is there also a notion of computational security in quantum cryptography (assuming a polynomial-time quantum adversary) where fully quantum information is being processed to begin with? Why does or doesn't this notion of security make sense?
There absolutely is. In fact, even in classical, there is the notion of computational security against polynomial time quantum adversaries. This is the whole point of post-quantum cryptography. This would let us keep using existing, classical, technology, but hopefully be secure against quantum-powered eavesdropping.
You are confusing two different things:
Quantum cryptography protocols that cannot be implemented classically, and having nothing to do with computational security;
Mathematical problems that can be solved both quantumly and classically, and here you can consider the computational security of algorithms.