Is a Qualified Digital Certificate backward compatible with existing certificate code?
TLDR: yes.

Qualified Digital Certificate is not a specification aiming at insuring interoperability. It purposely does not even specify a precise cryptosystem (see comment). Thus in principle there is no sure answer to that question.

However Qualified Digital Certificate only requires common X.509 features, modern Microsoft Cryptography API is flexible in what it accepts, and because it is common certificate issuers typically take care of going by whatever requirement it sets. Thus my bets are on: likely yes, and I'd be surprised if a little care in parametrization at key generation and procurement of the certificate did not make it possible.

Note: Qualified Digital Certificates does not imply Qualified Digital Signature, and there is a long way from the former to the later.