My parents never did this when we were infants, but my sister and I are both December babies very close to Christmas. Trying to have a party or parties for us was just not workable since everyone was either out of town with relatives or busy with their own holiday things. So, my dad came up with the idea of our "un-birthday". It was June 21, the first day of summer and we went to the park for bbq's, we had water balloon fights, swim parties. It was great fun. Up until the day he died a couple of years ago, he still sent me and my sister "un-birthday" cards.
Strangely, we don't do un-birthdays for our children. There was never really a reason to.
I guess if you think about it, every single day is a birthday.
If you want to celebrate it, do it. If you don't, no big deal. I think the first birthday is so fun because it's usually the first time kids get cake and candles and a big fuss.
Just my opinion.
Best wishes, whatever you choose.