To file as Head of Household, you must have a qualifying child or dependent.
There are conditions to satisfy in order to file as Head of household. You must:
Please read this TurboTax article for more information.
If you don't have a qualifying child or dependent, you must both file Single.
For Head of Household filing status you must have a qualifying person as a dependent and you must have provided over one-half of the cost to maintain the home.
See this TurboTax support FAQ for Head of Household - https://ttlc.intuit.com/turbotax-support/en-us/help-article/tax-filing-status/qualify-head-household-2021/L0NxUtVc2_US_en_US?uid=lu8x7m6p
Does one of you have a qualified dependent? You cannot file as HOH unless you have a qualified dependent.
If it was "just the two of you" ---no, neither one of you can file as HOH.
Am I Head of Household?
https://ttlc.intuit.com/questions/1894553-do-i-qualify-for-head-of-household
https://ttlc.intuit.com/questions/2900097-what-is-a-qualifying-person-for-head-of-household
If you qualify as Head of Household, when you enter your marital status (single or married filing separately) into MyInfo, and then enter your qualifying dependent, TurboTax will offer HOH as your filing status.