Q. Can she claim head of household for her son while I claim head of household for our daughter?
A. No. If you occupy the same home, only one of you can be HoH.
Q. Can we both claim head of household at the same time?
A. No. If you occupy the same home, only one of you can be HoH.
"We have a new dependent (daughter) together this year." If your daughter was born in 2020, she is not anybody's dependent for tax year 2019, nor is she a qualifying person for HoH, on your 2019 return,
Neither your girlfriend or her child can be your qualifying person for HoH, even if they qualify as a dependent (unlikely), because they are not related.
Your girlfriend cannot claim HoH, if you pay more of the household expenses than she does. Both of you may have to use Single filing status, for 2019
