Deriving the private key of old addresses that got reused and still have balance: eventually.
Deriving the private key of old address with no outgoing transactions: no.
The majority of "lost" coins fall in the latter category. Quantum computing can't bring back old coins just like it can't bring back your grandmother.
You forgot Satoshi's 1000000 bitcoins that have newer been moved, but still have their public keys visible.
Those addresses are difficult to protect from QC, because even if we implement a QC-resistant algorithm, those can't be protected or otherwise even Satoshi couldn't move them.
Actually once you soft fork to add the quantum resistant code, you might also add something to keep those coins locked forever. Something along the lines of: By the date the new code activates, coins not moved to a new wallet that supports it become locked. You can wait a year or four (next halving), etc. to give people plenty of time... Well we will see what core brings regarding this matter.