Father of my child has not contacted our child in over 30-day and has not given any type of financial support in 30-days...
Father filed for joint custody but has continued to not give child support for the child or have consistent contact.
I want to file for abandonment in Gwinnett do I have grounds to file for it?

If it has only been 30 days then there is nothing to file for. Is the child support through the state or is this something you both had agreed upon ?

If this is Georgia, then yes. GA calls failure to support a child as abandonment. If there is court ordered support that he has failed to pay, you can file charges of abandonment. On the other hand if there is pending litigation about custody, I would check with that court first.