If my son, his wife and newborn baby live with me, and I pay all household expenses, can I claim Head of Household?

Yes, they are filling a joint return and claiming her. 

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@lgh19 wrote:

You can Claim Head of Household if you pay more than 50% of all of thier support.  And they are letting you claim them as your dependents.  

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As was stated they are filing their own tax return and claiming the child on their return.  

The OP cannot claim the grandchild as a dependent since the child is being claimed by the parents.  The OP cannot claim the parents since they are filing a joint tax return.   Qualifying Relative rules apply -

To be a Qualifying Relative -

1. The person cannot be your qualifying child or the qualifying child of any other taxpayer. A child is not the qualifying child of any other taxpayer if the child's parent (or any other person for whom the child is defined as a qualifying child) is not required to file an income tax return or files an income tax return only to get a refund on income tax withheld.
2. The person either (a) must be related to you or (b) must live with you all year as a member of your household.
3. The person's gross income for the year must be less than $4,300 (social security does not count) in 2021
4. You must provide more than half of the person's total support for the year.
5. The person must be a U.S. citizen or a U.S., Canada, or Mexico resident for some part of the year.
6. The person must not file a joint return with their spouse.