Yes but your name must be on the child's Birth Certificate to get the initial 90 days Non Imm O visa and you can't get the 1 year extension of stay unless you've been through the Court and got legal custody of your child. You can apply for the 60 days Thai dependant relatives extension of stay.
Well yes, most offices only issue the 60 days visiting Thai dependants. Never seen an Immigration office issuing both the 30 and 60 days extention for the same visa/exemption/entry.