No, ECDSA and EC-Schnorr, as well as related schemes like EdDSA, all belong to the class of elliptic curve cryptography. Their security is based on the assumption that the EC discrete logarithm is unfeasibly hard to compute. This assumption is not true if a sufficiently strong general purpose quantum computer would exist.

Quantum resistant signature algorithms do exist, but they all rely on very large signatures - which may make them unfeasibly expensive for purposes like Bitcoin. Furthermore, much less research exists into features on top (like homomorphic derivation like BIP32 uses, or aggregation), making them effectively a step backwards in terms of functionality if we'd adopt that instead.

However, I am not very worried about this. Quantum Computing in general has a very long way to go before it comes even close to tackling problems like solving discrete logarithms for curves of our size. A general-purpose QC with several 1000s of qbits would be needed, and it is not even known whether it is physically possible to build such a computer.