QB Issue Resolution:

yes it is a business expense, I set up a franchise tax expense account and use that

but you can just a tax fee expense if you do not want to track specifically that tax

Resolution for Issue 'How do I post yearly LLC tax payment made to Franchise Tax Board?' available: Yes (Solved).
Source: Intuit Community forum.

How do I post yearly LLC tax payment made to Franchise Tax Board?: this issue or error code is a known issue in Quickbooks Online (QBO) and/or Quickbooks. Support for this issue is available either by self-service or paid support options. Experts are available to resolve your Quickbooks issue to ensure minimal downtime and continue running your business. First try to resolve the issue yourself by looking for a resolution described below. If it is a complex issue or you are unable to solve the issue, you may contact us by clicking here or by using other support options.

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