Yes, colouring the water could make the pool heat faster, though whether the colour you noticed has an significant effect is debatable.
Swimming pools exchange heat with their environment by conduction through their walls, evaporation at the surface and absorption of sunlight. I have little direct experience of swimming pool thermodynamics, but some Googling suggests that evaporation is the main way heat is lost, and (if it's sunny) absorption of sunlight is the main way heat is gained. Presumably heat conduction through the walls is slow enough to be small compared to the other two.
The sunlight hitting the pool is proportional to its area, so that isn't easily changed. However the proportion of the sunlight absorbed is equal to the incident sunlight minus the reflected sunlight, and you can change the reflectivity. For example if you paint the bottom and walls of the pool black they will heat faster than if you paint them white, and the walls will then heat the water faster.
Under normal circumstances water absorbs relatively little light, so sunlight heats the walls and the walls heat the water. However if you put black ink into the water the sunlight will heat the water directly. So the black ink would decrease the overall reflectivity of the pool just as painting the walls black would, and it would increase the rate of heating.
So, the brown colouring you noticed will (probably) decrease the overall reflectivity and thereby increase the rate of heating. However if, as you say, the bottom of the pool is still visible I doubt the effect is very great.