Whether you can get a residence permit is ultimately up to the local police bureau. In most cases a Q1 is not a prerequisite and they're more than happy to transition you from any visa onto an RP, if you are eligible for it.

In fact having a Q1 does not guarantee that you will get an RP from the Police Bureau, e.g., you may fail the health exam or the criteria for the RP in your jurisdiction differs from that of the Q1 visa. However, the Q1 is useful to verify that you have the right documents/authentication (e.g., marriage certificates/birth certificates must be authenticated by the Chinese Consulate/Embassy holding jurisdiction over the locality that issued them) that would be hard to obtain once you are in China.