Me and my boyfriend has lived together all year we have one child we have currently moved can my boyfriend claim head of household for the old address (6 months)
With claiming the child only as qualifying as head of house hold not as a dependent and I claim head of household with new address ( 6 months) with child qualifying me as head of household and I'm claiming the child as a dependent as well can we both qualify for head of household for 2 address with same child but only one of actually claiming him as a dependent?

I don't think so. Probably your best bet is for one of you to claim head of household and claim the child as a dependent (whichever one makes more money) and the other to file as single.

You BOTH cannot claim Head of Household; to do so is TAX FRAUD, and, in this case, it is an OBVIOUS case of tax fraud.

That said, if you both are the biological parents, one of you can file HoH and the other can file SINGLE. EITHER of you can claim the child as your dependent and still be within the letter of the law.

Another possibility: Your state MAY sanction common law marriage (a call to the local courthouse will answer that question). If so, you probably qualify, in which case you can file jointly and you BOTH claim the child.