My older daugher was bald as a cue ball until she was 2, but my little one was born with so much hair I couldn't believe it! We had to cut her hair at 3 months to get it out of her eyes. A 20 month old won't be a problem. My little trick is to find some of those mini-m&m's and feed them to her one at a time to keep her looking at you. That will help the stylist out if you can keep her still! Or, go to one of those kids salons. They have these nifty chairs for them to sit in that look like fire trucks or cars or whatever, and most of these places actually have a TV at every station. They'll put on Barney or Elmo (whoever your child's favorite is)- she won't even know her hair is being cut!
My daughter used to love getting her hair cut at that age because they would put little bows and even glitter in her hair - she thought she looked just like a "real" princess!
Good luck and remember your camera!