Since you are filing Schedule K-1 from the Massachusetts LLC, you would report only the income from the K-1 (if any). If the LLC does business in California, it may need to register and pay franchise taxes in California.
California does tax all income sources on an individual tax return. But a California resident would only file Form 568 if they filed Schedule C for a Single-Member LLC on their personal tax return. The pass-through income from Schedule K-1 doesn't require this form.
TurboTax will ask specific questions about the income you report. If there are state forms required to be included with your state tax return, they will be prepared and filed with your state return.
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