If you are not legally married then only one of you can claim the child on their tax return.
When two people are living in the same household and not married, only one person can file as Head of Household. The other would have to file as Single.
The one filing as HOH would have a qualifying person as a dependent and pay for over one-half the cost to maintain the household. Two people in the same household cannot contribute more than one-half each to maintain the household. One of the two is going to contribute at least $1 more than the other person.
See IRS Publication 501, Standard Deduction, and Filing Information for the Head of Household rules page 8 - https://www.irs.gov/pub/irs-pdf/p501.pdf#page=8
Keeping Up a Home
To qualify for head of household status, you must pay more than half of the cost of keeping up a home for the year. You can determine whether you paid more than half of the cost of keeping up a home by using Worksheet 1.
