Give me a water. Is this expression possible in conversation?

In your example, "a water" probably means a single serving of water.

If we use the indefinite article a with a normally uncountable noun, we usually do it to mean one of two things:

1. A certain kind of that noun.

2. A single instance or serving of that noun, such as a packaged container of it from a shop or a serving at a restaurant.

Here's an example for each case, respectively:

In Mexico I tasted a water unlike any I'd drunk before.

(To clarify, that's the past perfect used to mean that when I drank it, I had never drunk a water like that before.)

I was thirsty so I stopped at a store and bought a water.

You might also hear something like this:

I'm thirsty. I think I'll buy some juice. How much is a juice?

Again, that's referring to a single packaged juice, usually a single serving. If I'm talking about buying a large container of juice to keep in my fridge, I'll probably just refer to it as "juice" or "some juice". But if I were telling my wife (if I had one) to buy two large containers of juice, I might say: "When you're at the store, buy two juices." I could say "some juice", but it's understood when I say "two" that I want to units/containers, and this is using "juice" in a countable way.