Will the ECDSA with SHA-3 implementation be quantum-proof? I know, that ECDSA could be broken by quantum computers, but since there is post-quantum hash function SHA-3, does not this make the digital signature resistant to them?
No.  Anyone who can compute elliptic-curve discrete logs, such as an adversary with a quantum computer powerful enough to run Shor's algorithm, can immediately recover an ECDSA private key and forge arbitrary signatures of their choice.  It doesn't matter if you use SHA-3 or floss each day.