The only parent who has the automatic right to claim a child as a dependent is the parent where the child lives more than half the nights of the year (they do not have to be consecutive). Payment of child support is not a determining factor.
Assuming that is your ex, then she is the only parent who can claim the child as a dependent. If she is a taxpayer for 2018 (has income more than $12,000, or has lower income but files in order to claim any deductions or credits) then she must claim the child and she can't allow her boyfriend to claim the child even if she wants to.
If she is not a taxpayer for 2018, then either you or the boyfriend might be able to claim the child as an "other dependent". This would go to whomever provided more than half the child's total financial support for 2018. Your child support counts in your favor, but if the boyfriend provides a place to live, he gets to count 1/3 of his rent or the rental value of his home, as support for the child, plus 1/3 the utilities, 1/3 the household food, and so on.
However, the child must also live with the boyfriend for the entire calendar year. If the child lives with you part of the year, then the boyfriend can't claim the child, but if you don't provide more than half the child's support, then you can't claim the child either. In that case, everyone who provides more than 10% of the child's support must sign a "multiple support agreement" that specifies which person gets to claim the child. If no MSA, then no one can claim the child as a dependent.
A child who does not live with you can be claimed by you as a dependent for the child tax credit only if the other parent gives you a signed form 8332. Even with that form, a child that does not live with you never qualifies you for EIC or to file as head of household.
