First of all, we don't need 256 bit security. 128 bit security is fine; $2^{128}$ operations is already infeasible, especially when we're talking about security against quantum-cryptanalysis where the operations are more expensive than with a regular computer. Grover's algorithm is also known to be very resource intensive, requiring the emulation of the target algorithm using qubits.
Furthermore, while it is nice to have a bit of a margin when it comes to classical attacks, it's not likely that Grover's algorithm will be enhanced so that fewer qubits are required. Grover has been shown to be optimal for the "bruteforce" attempt against a cipher / key, so the number of logical qubits will at least remain the same. It is yet unclear how many qubits will be required for error conditions, so the number of required physical qubits is still unknown. That doesn't matter much as it won't alter the logical number of operation or qubits required.
DES is a block cipher, and triple DES consists of encryption, decryption and then encryption again of the block cipher. The mode of operation for the block cipher (e.g. CBC) is then performed on that stacked construction. ChaCha20 however is a stream cipher, not a block cipher, so you'd have to define what triple ChaCha20 would be like. ChaCha20 contains a PRF internally, and I guess that PRF can be tripled, but that's messing with the internal definition of the cipher.
That all said, quantum cryptanalysis is generally search, I'd say that yes, triple ChaCha20 would provide at least 256 bit security. However, as indicated, that's not required in any situation; defining or using triple-ChaCha20 would not make a system more secure in practice. Maybe triple AES-128 would make some sense if all you have is accelerated AES for that particular key size (I've seen some embedded processors that have those kind of limitations). Even AES-128 would be pretty hard to break though.
