Translation:I have a very good relationship with him.
Verbindung is connection. The best example of its meaning is - Internetverbindung (Internet connection). Or connections like acquaintanceship.
On the other side, we have Verhältnis which means relationship between two things, units (people, animals, or hemical stuff like moleculs) Verhältnis also can be used in a term when you want to describe two people who have just sexual relation.
Beziehung is always relation between two living creatures and we use it for romantic relationship also, and usually people get it like that, if you do not precisely describe which kind of relation it is.
Unlike in many other European languages, "a" and "one" are two distinct words with distinct uses in English. You only use "one" when you're focusing on the number, when you want to contrast it with two or more. Saying "I have one very good connection with him" sounds very weird, like as opposed to two or three very good connections with him???
It ('one') can also emphasize the connection part strongly. Think of another sentence that goes: That's one fine Scotch you're pouring there. Emphasis on the fine Scotch. You can also say - That's a fine Scotch you're pouring there. That means the same thing, but it's more casual and no extra emphasis. If you're not trying to place extra emphasis, use 'a' rather than 'one' in these situations.
Verhältnis is neutral/akkusativ following an indefinite article (ein). It would be 'gute' following the definite article (das), but that wouldn't make sense for this sentence. See here (I've found this webpage very useful!) http://esl.fis.edu/learners/fis/german/kasus/caseTables.htm
There seems to be a strange quirk to 'gut' whereby it can't be used to describe a romantic relatinship. In an earlier lesson, there was a sentence like "du bist ein guter Freund," and "You are a good boyfriend" wasn't accepted as one wouldn't describe a boyfriend as 'gut.' Can a native speaker correct me if I'm wrong?