Most attacks now on classical computers don't actually break the encryption, they trick the systems / communication protocols into using it in a weak way, or into exposing information via side channels or directly (via exploits like buffer overflows).
Or they trick humans into doing something (social engineering).
I.e. currently you don't attack the crypto itself (because things like AES or RSA are very well tested), you attack the system built around it and the people using it.
All of these avenues of attack will sill be present when computers communicate via quantum encryption. However, with quantum encryption theoretically giving perfect security instead of just computational security, tricking a system into weakening its encryption (by using weak keys or wrong keys or keys you already know) shouldn't be a problem.
Possibly there will be weaknesses that systems need to avoid in quantum crypto, especially practical implementations that work over imperfect channels.
TL:DR: When quantum computers can practically attack RSA and the world switches over to quantum crypto for communication without a pre-shared secret, we'll be back in the same situation we are now: the crypto itself is not the weakest link.