The preferred approach for a commercial product would be to contract some cable company to make exactly what you want. There are lots of vendors that do this.
You could try this: 90 Ohm FFC. But I don't know much about the vendor.
You could also look at off the shelf LVDS FFCs or shielded FFCs. There are quite a few out there with 100 ohm differential impedance.
Another option would be to build your own flex PCB with controlled impedance.
...all that being said...
If you can't find impedance controlled FFC and this is for a non-commercial product, you'll probably be okay just using standard FFC. Here's an ... interesting article ... about the subject from Microchip. I couldn't quite figure out what the bottom line was.
Consider that on FR4, a 120mm distance would have a propagation delay of about 1ns, so anything with a rise time of 5ns+ would probably be okay. Full speed USB would probably be okay here at 12Mbps (83ns /bit), but high speed USB would be really stretching it at just 2 ns/bit. Your propagation speed in an FFC will be different from FR4, but this is just an estimate. You are also probably going to have some nasty EMI if you have impedance discontinuities so it may not be an acceptable approach for a commercial product.