My grandmother says to shave my 1 year olds head in order to make her hair thicker.  is there any truth to this?

NO NO NO NO NO NO NO. Hairs are thicker at their base, and thin out, taper out and get softer at their ends. So while the hair may seem thicker as the initial stubble grows out, ultimately with time, the same lovely child's hair will gradually reappear. Hair is important to protect the delicate baby's scalp from damaging sun rays, so that makes shaving a baby bald very unwise. I am glad that you came to healthtap!
Old Wive's Tale-1. This is strictly an old wive's tale. See dr. Fein's answer for the reason.