Can two people claim head of household?

I assume you are talking about an unmarried couple living together who have a child or children living with them, who meet all the requirements for Head of household (HOH) and the dependency exemption.

Yes, they could potentially both file as HOH, but only if they each have a qualifying child who lives with them and all other tests are met. So if you only have one child living with you, than only one qualifies for HOH. If there are two children living with you, than you can each claim one of the children for HOH purposes and for the dependency exemption.