I bought a pillow for my daughter (now 16 months) when she was around 12 months of age. I bought it from a small baby shop we have in the area but have been told Target and similar shops sell them a well. They are smaller than an adult size pillow, not as thick and apparently breathable (if u know what I mean). Emma sleeps on hers every night without fail.
You can always buy one and try it and if bubs doesnt like it put it away and try it again later.
I gave my 19 month old a pillow a week or two ago. I thought it might help her stay under the covers and keep her in the "right" position, rather than moving all around the cot. It's worked really well, and I think she'll be ready for her big bed before long.
My son's would never sleep with a pillow in he's cot. He would throw it straight out. But since he's been in the bed for 2 weeks now he sleeps under the covers with a pillow and ALL NIGHT. (Nicholas's 2 1/2years old).