No, only one of you can be the custodial parent and claim benefits related to a child living in your home. Therefore, only one of you should include the child on your tax return. The child could only have lived in one of your homes for the entire year and you must decide who will claim the child. If you each have the child for the same amount of time during the year, then neither of you will qualify to claim the child.

The IRS gives the dependency exemption to the custodial parent (the parent who had the child longer during the year) unless that parent allows the noncustodial parent to claim the child. However, there is a special IRS rule that allows divorced or separated parents to split tax benefits for the same child. Under this rule, the custodial parent is the parent who had the child for the longer portion of the year. 

Note: You do not both include the child on your tax returns unless the noncustodial parent claims the child as a dependent as described in the IRS link above.