2010 Toyota Corolla - Water Pump Replacement

Dear Friends,

My niece took our Toyota Corolla 2010 to a neighborhood auto shop for oil change and they informed her that the Water Pump is leaking and needs to be replaced. Else the engine will over heat and blow up. We live in Socal so we have extreme heat some times.

Later drove the car for about 10 miles and checked underneath the car if water was leaking and did not see any leakages. What ./ where exactly I should be looking at to find if the pump is really leaking?

The quote was $ 475 to replace the water pump. Replacement will be original Toyota pump. I am willing to spend that if it is really an issue with the pump leaking. How to check if the solution is pump replacement and not replacing the leaking gasket /belt?

Thanks in advance
Usually you will see pink fluid splattered across the underside of the hood from it. Look to see if you can see any signs of that or have the place that is telling this to you to show you. I've heard it's very simple to replace so $475 sounds expensive.
I'd ask that shop to show you the leak or explain how they determined the water pump is leaking, but before you go back there, make sure your radiator overflow/fill tank has liquid it in. Normally it is pink in color and should be about half full. There are marks on the tank showing the preferred fluid level. You can also start the car (make sure the air conditioner is fully OFF at all times) and watch under the passenger side of the engine area. If you see dripping, it's either from the puddle you just drove through, a running air conditioner, or a leak. You don't want to drive the car if the overflow/fill tank is empty as it feeds coolant (antifreeze) to the radiator and engine. It is normal for the level in this tank to go up a bit and down a bit depending on the engine temp, but it should never be empty.
If you do need to change the water pump, get more than one price - shop prices can vary a lot. Even dealer prices vary from dealer to dealer.
I would get a second or third opinion on this. Not uncommon for oil change shops to drum up business by stating a car needs this or that. I many cases the customer is duped into this and they don't even replace anything, just charge you for it. I have not replace a water pump on a corolla yet but it would seem to be straight forward to do, I think time would be the biggest factor but that price seems high to me.