When Jesus was on the cross, and said "My God, My God, why hast thou forsaken me?", his words may well have been spoken in Hebrew or in Aramaic. This is recorded in both Matthew and Mark, with different spellings in each.
And about the ninth hour Jesus cried with a loud voice, saying, Eli,
Eli, lama sabachthani? that is to say, My God, my God, why hast thou
forsaken me? (Matthew 27:46, KJV)
And at the ninth hour Jesus cried with a loud voice, saying, Eloi,
Eloi, lama sabachthani? which is, being interpreted, My God, my God,
why hast thou forsaken me? (Mark 15:34, KJV)
Matthew has the original words as if transliterated from their Hebrew spelling; Mark spells them as if transliterated from Aramaic; hence the slight difference in their English transliterated spellings. Both are writing Greek characters, but these are transliterations. This can be seen by the very fact that they are afterward "interpreted" for us--where it is written in proper Greek.
Some other statements in the New Testament known to have come from Aramaic or Hebrew include:
- Matthew 5:22 -- "Raca" (Strong's says "of Chaldee origin".)
- Mark 5:41 -- "Talitha cumi; which is, being interpreted, Damsel, I say unto thee, arise." (The "Talitha cumi" is from Aramaic.)
- Mark 7:34 -- "Ephphatha, that is, Be opened." ("Ephphatha" is from Aramaic.)
- Mark 11:9 -- "Hosanna" (of Hebrew origin)
- Mark 14:36 -- "Abba, Father" ("Abba" is "of Chaldee origin"; "Ab" is father in Hebrew)
- John 20:16 -- "Rabboni; which is to say, Master" ("Rabboni" is "of Chaldee origin")
- Romans 8:15 -- "Abba, Father" ("Abba" is "of Chaldee origin")
- 1 Cor. 16:22 -- "Maranatha" ("of Chaldee origin")
- Gal. 4:6 -- "Abba, Father" ("Abba" is "of Chaldee origin")
NOTE: Chaldee and Syriac are essentially other names for Aramaic, with time/evolution of the language being the only distinction among them.
Respecting the beliefs of Seventh-day Adventists, I am unaware of any claiming the New Testament was written in Aramaic, as the question supposes, but some may. What is demonstrable, however, is that Ellen White, a co-founder of the church, had this to say about the Bible's origination among her travel notes.
"Here [Bale, Switzerland] Zwingli received his early education; here
Erasmus published the New Testament which he had translated from the
original Greek into Latin; here Frobenius, the celebrated printer,
published the writings of Luther, and in a short time spread them in
France, Spain, Italy, and England; and here, too, John Foxe spent a
portion of his exile in getting some of his books through the press."
(Ellen White, Review and Herald, 1885).
That should settle the question for Seventh-day Adventists. However, because that statement is not found among her major books (notably it is not in "The Great Controversy"), it may not have been seen by all, and those ignorant of it may have developed other views.