No, a part-year resident of California needs to report only California source income in the time period that he/she is a non-resident (which is usually zero).
Please see the following:
Part-year residents of California - Taxed on all income received while a resident and only on income from California sources while a nonresident. (see https://www.ftb.ca.gov/individuals/fileRtn/Nonresidents-Part-Year-Residents.shtml#how_taxed).
Part-year California residents often find that TurboTax adds back all the HSA contributions, even ones made in another state.
There is no good way for TurboTax to know in which state the contributions were made in, so the taxpayer needs to manually adjust the California state income to remove the HSA contributions that were added back while the taxpayer was not in California.
For example, if the taxpayer lived 6 months in California and 6 months in Texas, and made $3,000 in HSA contributions evenly distributed over the year, then the taxpayer should adjust the California state income by $1,500 to remove one half of the total HSA addback that was done to California state income by TurboTax.
Another example is that a taxpayer who lives in California but moved to Texas on July 1st, but all the HSA contributions were made while in Texas. TurboTax by default adds the entire HSA contribution back to California state income. Because none of the HSA contributions were actually made while the taxpayer was a California resident, the whole HSA contribution for the year needs to be backed out of California income.
***To make the CA adjustment***
Go to State Returns, and navigate to your California return.
In Income and adjustments, proceed through the interview. You may see a screen announcing that HSA contributions are treated differently in California. Just hit Continue.
You will notice on the main page ("Here's the income that California handles differently"), the first line item is (likely to be) "Health Savings Account (HSA) Contributions". Here TurboTax notes that the amount of your HSA contribution has been added back to the California return.
NOTE, despite the Edit button, you can't change this here.
Scroll down to Miscellaneous Adjustments on this screen. Click Start for Other Adjustments to Income.
Enter in the left column "adjustment for out-of-state HSA contributions". Enter in the middle column (i.e., a subtraction) the dollar amount of HSA contributions made out-of-state. This will be subtracted from your California state income.
Make a note on your copy of your state tax return (because, of course, you are going to save a copy, right?) that you made this adjustment because TurboTax added back all the HSA contributions (even ones made while a non-resident), and you needed to counteract this. This is in case you ever get a letter from the state asking about this adjustment.