No, you don't pay the minimum franchise fee for your first tax year.
"Newly Incorporated or
Qualified Corporations
To compute the tax for your corporation’s first tax year,
multiply your California net income by the appropriate
tax rate. Your first tax year is not subject to the minimum
franchise tax.
After the first year, your tax is the larger of your California
net income multiplied by the appropriate tax rate or the
minimum franchise tax."
- FTB Publication 1060
