This TurboTax reference explains in detail the qualifications for filing Head of Household: https://turbotax.intuit.com/tax-tips/family/guide-to-filing-taxes-as-head-of-household/L4Nx6DYu9
Theoretically you could each file HOH if two different qualifying dependents were involved. But you cannot each use the same child as your "qualifying dependent."
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
