Unfortunately we can't answer that question yet. People often drastically overstate the theoretical power of quantum computers. Since the actual power of quantum computers is currently zero (they don't exist), the actual threat is zero and it's hard to reason about how big the ones that might it exist may be because we don't know how to build them.
That said— A very large and fast true quantum computer of a scale which may turn out to not be physically possible could create problems for our ECDSA signatures.
The design of Bitcoin minimizes the exposure there: If you use bitcoin addresses only once (as is intended for privacy reasons) then your ECDSA public key is only exposed right when you spend, which is just minutes before an ECDSA attack would no longer be effective— so an attacker wouldn't just have to be able to compromise ECDSA they'd have to do it fast. It's likely that once ECDSA attacks became feasible they would be slow for a long time.
(E.g. even though RSA-512 is crackable (enough so that crazy over computing powered people like me have done it at home) Bitcoin would not be fatally insecure right now if we used RSA-512)
Secondly, our scripting system allows for some kinds of backwards compatible changes. If QC ECDSA attacks started becoming threatening looking we could extend Bitcoin with a resistant signature algorithm (like lamport) and create transactions which require both ECC and Lamport keys. Old nodes would validate only the ECC key and ignore the lamport, new nodes would validate both.
(The hash functions in bitcoin are probably secure— at least to QC specific threats— QC only provides a sqrt(n) for black box non-linear inversion— so a 256 bit hash has the same security under QC that as 128 bit hash has on classical computer, which is sufficient. This does imply that if miners got QC's which did as many operations per second as their classic computing hardware (a crazy assumption but whatever) then the difficulty would square— e.g. we'd go from difficulty 1,000,000 to 1,000,000,000,000)