Am I considered head of household if I’m not married but my fiancé have lived together for the full taxable year? I have provided half the financial support for the home, her, and her son. W

If "her" child is not also your child then no.    You must have a closely related dependent that lives with you to qualify for HOH.

You cannot file as Head of Household because you don't have a related person living with you to qualify you to file with that status. You will have to file Single.

See these requirements:
1. You are not married or “considered unmarried” on the last day of the year. You are considered unmarried if you lived apart from your spouse the entire last 6 months of the year.
2. If "considered unmarried" you file a separate return from your spouse.
3. You paid more than half the cost of keeping up your home for the year.
4. You have a qualifying child (whether or not you claim the dependency exemption) or you claim a dependency exemption for a “qualifying person”  who is related to you and who lived with you in the home for more than half the year. An exception is a parent does not have to live with you to be a qualifying person.