Since Christianity stems from Judaism did the Jews believe that their Messiah and their God would be the same person as described in the Trinity?
No, neither the Jews (those left after the northern exile) nor the entirety of the Hebrews (all those of the faith before the northern exile) held to any concept of a trinity. The ancient egyptians had a trinity, but not the Hebrews. In fact, they held to a strict monotheism as taught by scripture.
Deuteronomy 6:4 The Scriptures 1998+ (4) Hear, O Yisra’el: יהוה our Elohim, יהוה is one!
Fortunately for the purposes of our discussion, the verse of the translation above is a word for word and in the same order as that of the original Hebrew. This was pretty much the creed of the faithful and the creed that our Messiah Yahushua re-iterated in Mark 12:29. The above is extremely important due to the grammatical construction of the sentence in Hebrew. For those of you who don't know, in order to express something as singular, you place the numerical designation after the noun. But in order to express something as plural, you place the numerical designation before the noun. In English we always place the numerical designation before the noun so it's not as obvious to us when we read it. But in Hebrew if you wanted to say "3 o'clock" you would say "hour three". It's the third hour, but it's still a singular hour. However, if you wanted to say "three hours" then you would simply say "three hours" to indicate plurality.
This brings us back to the verse above. Here we clearly see that the numerical designation is AFTER Yahuweh (aka God). Take a look at the original Hebrew for further verification. We can clearly see how the Hebrews, then the Jews, and then the Christians (at least up until the Catholic councils) would view Yahuweh (aka God) as one as opposed to seeing Yahuweh (aka God) as three as is stated in the theory of the trinity.
And also, no, the Hebrews and then just the Jews (after the northern exile) would not have thought of Yahuweh (aka God) as becoming human and dieing. The scriptures clearly indicate that Yahuweh is eternal and cannot die (Deuteronomy 23:40, Psalm 90:2, Psalm 102:27) and the Hebrews would also have read many times as they went over and over the Torah, how Yahuweh is not a man nor a son of man:
Numbers 23:19 The Scriptures 1998+ (19) “El is not a man, to lie; nor a son of man, to repent! Has He said, and would He not do it; or spoken, and would not confirm it?
So to answer your question directly, no, the Hebrews, then the Jews (after the northern exile), and then the early Christians until the Catholic councils would not have imagined that Yahuweh (aka God) would come to earth as the Messiah.