Urinating 20 times a day at 4.5 months falls within the range of normal--especially if you are using diapers full-time. Babies at that age will release their urine in small amounts if they are not given the opportunity to relieve themselves away from their body (ie in the potty/toilet). Often they will go in little spurts, because their bladders are small and in some cases, they do not want to get themselves wet, so let out as little as possible to relieve the pressure. My daughter would not fully relieve herself in a diaper, she would always stop right after she started--on of the major reasons we did EC (elimination communication) with her. If we put her on the potty she'd pee till empty and be dry for a couple of hours, otherwise, she'd pee every 15-20 minutes :-/
More than 2-3 times an hour all day long is a sign something may be wrong though, so don't totally discount your baby's urination patterns--in this case look for UTI, yeast, food/chemical/environmental intolerance, constipation or genital irritation.
*Edited to reply to update*
"When carry her and walk she stops that sound. If she is ill, I don't think she will stop it when carrying."
If your child is having stomach trouble, it is very likely that she will stop "moaning" when she is carried. The change in position and the warmth of body-to-body contact will comfort her. Definitely go speak with your doctor about the moaning and what may be bothering her.