Can I give my 2 year old melatonin?

Maybe. Though "natural" melatonin has not been studied for normal kids. It is mostly used in autistic children with sleep disturbance or other significant behavioral issues. It is most likely safe, but if you can avoid it then do so until at a safer age of over 3 yrs old. If u give it, start with 0.5 mg and increase slowly to max of 2 mg. Plan to use only for 2-3 wks while u stablish a sleep routine.
Yes. Melatonin can be given to two year olds. The question though is why do you want to? Melatonin is a lousy sleeping pill; it is a sleep cycle regulator, not a medicine used to induce sleep. It is useful for helping to change bedtimes or for jet lag etc. If you're trying to put your child to sleep, have you tried warm milk? Chamomile? Baths before bedtime? Bedtime stories?