Short answer, for federal tax returns is NO:

Under federal law, loan forgiveness generally counts as taxable income, and states almost invariably incorporate this provision into their own codes. The Coronavirus Aid, Relief, and Economic Security (CARES) Act, however, expressly excludes the forgiveness of small business loans under the Paycheck Protection Program (PPP) from this provision. Since states generally follow federal treatment of debt discharge, they would be expected to incorporate this exception as well—but only if they conform to the most recent version of the Internal Revenue Code (IRC), which includes the exception.

California: To be determined as of May 14, 2020. California will have to pass legislation to conform to this part of federal tax law.

Recap: You can take expenses (e.g. payroll) funded by the PPP loan subsequently forgiven and don’t include the income of the forgiven loan on your federal tax returns. (Note that your accounting records will need to show the forgiven amount clearly in “other income” so this item can be segregated when preparing the income tax return).