No it's not true. BUT... the reason it exists is because it's "sort of" true. LOL... it's because the ends break off, and they break of higher than if you kept it trimmed.
Just as an example... when I'm getting my hair trimmed every 2-3 months, I grow an average of 15 inches of hair a year. (I have fast growing hair). When I'm broke or busy, and not trimming... I grow about 5 inches of hair a year). Same diet, same exercise, same follicles... but one time I'm trimming before it can start breaking, and the other, they just break off. Kind of like fingernails. When I trim them, they don't snap off below the cuticle.
I've had 1/4 inch hair and waist length hair. ((I get ticked at my hair about once ever 10 years and shave my head)).
Children's hair is all spiffy-glossy because a) growth hormone that is saturating their system, and b) because the follicles are still developing. Her hair will change gradually as she grows, in part due to follicular maturity and in part due to the different types of hormones in her system. Cutting it will have no effect on her hormones. So the texture will stay the same. The ENDS won't be wispy anymore, unless she has curly hair (which tends to break a lot more than straight hair, just because of the hydrogen bonds in the proteins)... but again... that's a breakage thing.