Is ECDSA secure from quantum computers? Will quantum computers be able to calculate it's private keys from the knowledge of the public key once there are working quantum computers?
Short answer: Yes.
Elliptic curve cryptography is vulnerable to a modified Shor's algorithm for solving the discrete logarithm problem on elliptic curves.
To quote from pqcrypto.org: "Imagine that it's fifteen years from now. Somebody announces that he's built a large quantum computer. RSA is dead. DSA is dead. Elliptic curves, hyperelliptic curves, class groups, whatever, dead, dead, dead." Worth a visit to their site to understand what crypto looks like after the (hypothetical) success of quantum computing.