In Italy we usually refer to the names of the notes using the solmisation Do Re Mi Fa Sol La Si (the Ti name as a variant for Si is very uncommon in Italy). Jazz musicians probably are more accustomed to the letter names for notes than, say, folk singers or classical musicians, and may use both standards. When it comes to violin and you want to refer to the fact of playing on the G string (I guess that's what the instruction means, i could be wrong), I think an Italian would probably say "sul sol".
Your confusion is totally understandable, and I'd note that it can get even worse! For example, if I want to refer to Do/C, and pronounce it as C, it sounds just like the Italian note Si/B.
Unfortunately, I don't think there's a real rule on how to behave in this case: the existence of things like "sul G" is due to the fact that Italian is the language of classical musical annotations, but the letter-names A, B, C etc are way more common worldwide, so we ended up with this chimaera of sort.
After re-reading your question I can see that you were also interested in how to actually pronounce those terms. For both cases, I'd say something like this
"Sul G" is a bit like english "sool gee", while "Sul sol" would be something like "sool sol" where the last 'o' is quite open. Using IPA we can give more precise descriptions: those would be, in my opinion (and according to my accent) [sul dʒi] and [sul sɔl] respectively.