I don't know about the particulars of the app, but at the end of the day, the only data stored in the SQRL app is the private key. All website specific keys are derived from a combination of the domain of that website and of the private key. So there is no need for a sync mechanism per se, outside of originally sharing the private key.
That being said storing that key on your (encrypted) mobile only is what protects you.
There is an option to maintain the state of which websites were used to authenticate, but that's only to reset your access should your private key be compromised (or mobile stolen). Then your SQRL app should sync the list of websites with some central location. This would ensure that the day you think the key is compromised, you can reset all access to all websites with a new private key automatically (this is part of the SQRL protocol).
On 1Password, I don't see why not. Gibson made the protocol open source and free.