If your child is fussy and doesn’t seem to want to go to sleep while lying flat, you can give them the same sort of elevation they’d get from a pillow by placing a pillow or a towel underneath the mattress of their crib. That will perhaps make them more comfortable, but won’t be something they could roll over onto, or pull onto their head.
After 2 Years
From two years your child will have developed enough to be able to avoid the possibility of suffocation, and will also be bugging you for a pillow because they’ll be entering the terrible twos. They’ll see that everyone else in the household is sleeping with a pillow and they’ll probably throw a tantrum if they don’t get their own.
It can be a nice bonding moment to give them their first pillow, and if you can work it out so that they associate their pillow with nap time, you’ll feel like SuperMom. This is a nice side effect of holding off on giving your baby a pillow until they’re 2 or older.