No, the sentence you've written isn't quite right. I think you may have misheard this. It looks like a combination of two sentences: 'Do you mind grabbing me a water?' and 'Could you grab me a water?'. Another possibility is that the speaker might have said something like this: 'Do you mind...? If you could grab me a water....?' These are two half sentences. The full sentences might be something like this: 'Do you mind [doing me favour] ? If you could grab me a water [ that would be great] .' In natural, native-speaker conversations, we often use unfinished sentences of this type. In fact, 'If you could ....' and 'If you could just....' are fairly common ways of politely asking someone to do something for you. If you say them with a rising intonation, it's clear from the context that you are making a request. A few extra language points: 1. As Karina says, if we say 'a water', we mean 'a [ bottle of ] water'. 2. As Ray says, 'grab' is informal way of saying 'get' or even 'buy'. It suggests that the other person is on his or her way to the fridge, a bar or a store to get something for themselves, and the speaker is asking if he or she would mind picking up some water at the same time. 3. This is the same in all varieties of English. It makes no difference whether the speaker is in the US, the UK, or anywhere else.

at times "grab me a...." may mean "get me a..." or even "buy me a ...."

it sounds a little unnatural for UK "do you mind grabbing me a glass/cup/mug/beaker of water?" please. "could you grab me a glass/cup/mug/beaker of water?" please

It is a very casual way of asking someone for water. It is implied that it's a bottle of water, not a glass or a cup.