getting the/a garden view room

No. "The" indicates shared information - ie information known to both speaker and listener. So, eg if you knew that the hotel only had one room with a sea view (an obviously the staff would know that too) you might say :
I'd like to have the room with the sea view
In other words, you're saying "I know that you know which room I'm talking about".

But when there are lots of rooms with a sea view, so you're not referring to a specific one, and therefore there is no shared info, you'd say :
I'd like a room with a sea view.

Thank you- I'm sure your information is correct, but my line of thought went like this:
there are several possible views in the hotel- a garden view, a sea view, etc. so isn't it possible to specify the type of view one 'd like to have? ex. "I'd like a room with the sea view" ?
Thanks in advance

No - because there isn't "just one" sea view unless there's just one room overlooking the sea. By starting the sentence "I'd like a room..." you are presuming there are a number of rooms like that - and each one will have a slightly different view depending on it's position. So there is not one sea view from the hotel, but lots of different ones depending on whether you are on the first or tenth floor, the left of the building or the right. So what you are saying is "I'd like a room (it doesn't matter which one) with a sea view( it doesn't matter which one). Only if there was only one room overlooking the sea (and therefore only one view) could you say "I'd like the room with the sea view."

Thanks susan., your answers are so inspiring. 🙂