California discontinued the smog exemption for engine sizes 50 Cu and under. All 1975 and newer gasoline and diesel powered USED vehicles, with very limited exceptions, must now undergo a smog check during initial California registration and biennially. In the case of your Suzuki Cappuccino, yes it will need to receive a CARB Laboratory Test and Referee Center smog inspection. Once these certifications are obtained, registering the vehicle in California would be the same as if having brought the vehicle in from out-of-state. It's the lab and referee inspection that makes the process a bit difficult and costly, no to mention customs and importing fees.
Here's the link directly to the California Air Resources Board with some more details.
Registering an NON-USA, out-of-country, or "Grey Market" car in California
For more California vehicle registration information we recommend visiting NeedTags.com