My mom thinks she is constipated and does not remember when she has a bowel movement. Is there a solution?

Wow. We went through the same with our mother. She was obsessed with moving her bowels, but with the dementia, she couldn't remember going and then would say she was constipated and take milk of magnesia sometimes many times a day. The white board for your mother sounds like a good idea if she is competent enough to understand how to use it.
Mom also had anxiety attacks in the morning (more like panic attacks because a few times she threw up and lost control of her bowels and said she was dying) I'm guessing this is part of the aging process for some women. The doctor gave her clonzepam which she carried with her and she was instructed to take one at bedtime to keep it from happening in the night.

Is she capable of keeping a calendar after she has actually had a BM? She could write the time on the day and the letters BM?

I'd get a little whiteboard at the dollar store or a little calendar and mark on it every time she has a bowel movement. Keep it in the bathroom. Then she can just look at it to see she's already gone that day or you can show it to her to remind her that she's already gone today or yesterday. Make a big deal about marking it down each time.

I also think the anti-anxiety med at bedtime is a good idea as well.

But, MyThreeSons, she isn't really constipated! :D LOL she just thinks she is.

For constipation or slow bowels, V-8 helps my dad & is nutritional, too. I found that rhubarb juice is a great stool softener, too. After washing stalks, break them in pieces and oil with water until tender. Put into blender with sugar to taste to break up strings and to pulvertize.

Some meds like prednisone & pain killers can cause slow bowel action. If not overdone, either of these juices can help

I wonder if it wouldn't be easier all the way around to just go along with her. Keep on hand some kind of medicine-looking candy (mints, perhaps) and when she wants you to give her something, hand her two, and tell her she needs to dink a full glass of water after taking them. No harm done, and she might relax a little.

Anxiety is pretty common in the early stages of dementia, because she knows she is losing function and she will worry over it. Consider asking the MD for an anti-anxiety medication in a low dose, to be given at bedtime. That way she can get a good night's sleep and be less anxious in the morning.