In essence, the stimulus checks act as an advance of your 2020 income tax refund. This means when you prepare your 2020 income tax return, there will be a line to include the section 6428 credit (line 30 on the 2020 form 1040). The credit on your 2020 return is subtracted by any amount received as a stimulus check in 2020. If the amount you received as a stimulus check is less than the credit you are due, the difference will be included as part of your 2020 refund.
On your 2020 federal tax return you will be able to enter the amount of the payment that you are eligible to receive on the Form 1040 Line 30.
You should get the 1st and 2nd Stimulus payments for him on your 2020 tax return. Did you already file? It should be on 1040 line 30 if you qualify for the $1,100 (500+600).
Did he show up as a child and get the Child Tax Credit? If not you may need to amend. Did you say he lived with you All year? A child born (even on Dec 31) is considered to live with you the whole year. So pick the Whole Year at the top of the drop down box.
Here's info on the New 2021 3rd Stimulus payment, Unemployment and Child Tax Credit (scroll way down)
The new Child Tax Credit is for 2021, next year, when you file 2021 tax. It also raises the cut off age from 17 to 18 and makes the credit $3,600 for a child under age 6.