Hello, my was child born in January 2021. I heard they are eligible for the second stimulus payment of $600, which I haven't received. How can I go about claiming it?

The child is not eligible for the 2nd stimulus payment since he was born in 2021.  The second stimulus was only for a child born in 2020 or earlier. 

Your child will be eligible for the 3rd stimulus payment of $1,400 on your 2021 tax return.  Answer the stimulus question in the TurboTax program after you have completed the federal tax return in the Federal Review section.