The simple answer to this question is, yes, it can do - but not always. It is normal for mothers to lose some blood as the baby is born; so this can colour the water. I think the photo above is really beautiful and shows the kind of colour change you could expect. However, I have seen pool birth photos and attended a pool birth where there was no colour change to the water at all. As with all things to do with birth, it is highly individual.
It is less likely that the water colour will change during the labour. If the mother's waters have broken and the baby passes meconium (the black, tarry poo that newborns have in the first 3 or 4 days of life); this may stain the waters a little. Mother's would usually come out of the pool at this point as the baby may be signalling that they need some help and the midwife will do some extra checks to find out. Of course, it is possible to bleed during labour too, but this is more unusual. This would not be considered normal and, once again, mothers would often come out of the pool to have some extra checks at that point.I remember being a birth partner at a pool birth. The mother was in very strong established labour. She was telling me that she needed a wee; I asked the midwife what to do and she just encouraged the mum to wee in the pool. She reminded her that urine is sterile as it's released; that it would be a small volume of urine because the mother had been using the loo through the labour before she got in the pool and had not had much to drink since; and also that the pool was a very large amount of water that would easily dilute the small amount of urine to be negligible.
So if birth partners fancy hopping in the pool with mum to give some oxytocin boosting cuddles and massage; they'll usually be quite safe during labour but things might change regarding the water in the pool as the baby is born.