If I were lucky and won the lottery and I opted to take payments over
20 years at a specific interest rate and I decided to take my first
payment right away rather than waiting one year would I still earn
interest on the right away payment or would it be best if waited and
took the payment a year later?

Hi Baseball2,
No, you would not earn interest on the payment you received
immediately.  Like in any other situation dealing with interest, you
can only earn it if the other party has the use of the funds during
that time.
Whether it would be better to take that payment immediately instead of
waiting for a year depends on what you can do with the money.   If you
can invest it at a higher rate of return, you would probably prefer to
do so.

Myoarin