I have a 3 year old. Biological dad is in another country. He hasn't seen the child since birth though he knew about the child. He does not give any financial support nor contact us. I am now married for almost 2 years. My husband treats my child as his own. Since there is no father listed in my child's BC, can I add my husband's name as the father? My child was born in California.

Yes it would be illegal to do that.
Its also imoral too.

I have a 3 year old. Biological dad is in another country. He hasnt seen the child since birth though he knew about the child. He does not give any financial support nor contact us. I am now married for almost 2 years. My husband treats my child as his own. Since there is no father listed in my child's BC, can i add my husband's name as the father? My child was born in California.

No!
If the child was born before your marriage you have no legal resources to claim the child was born in marriage.

I'm not here to judge whether it's moral either; however, as someone who had this happen to her I'd love to know whether you plan to tell the child at some point? I found out when I was 26 that my dad (who is on my BC) is not my biological father. I wish I'd known early on, but I understand how hard it must have been for the parents to find "the right time.