Greek was the lingua franca of the eastern Mediterranean when the NT was written. It had been for a long time by then.
Multiple texts in the Apocrypha were written in Greek. 4 Maccabees, for instance, was written in Greek. 4 Maccabees recounts the martyrdom of 7 brothers and a mother by Antiochus who were tortured and then executed. Are you going to argue that because it was written in Greek that those martyrs who were ethnic rebels against Seleucid Greek control were themsleves Greek?
Jesus spoke in Aramaic and there are multiple Aramaisms in the Greek New Testament.
Here's an example from John 20:16:
λέγει αὐτῇ Ἰησοῦς· Μαριάμ. στραφεῖσα ἐκείνη λέγει αὐτῷ Ἑβραϊστί· Ραββουνι ὃ λέγεται Διδάσκαλε.
My translation (but cross reference with a bible if you want):
Jesus spoke to her, "Mary." Having turned she [lit. "that woman"] spoke to him in Aramaic [lit. "in Hebrew"], "Rabbouni" which means "teacher."
The word Hebrew in the Greek NT is translated as Aramaic in English because it was the demotic language spoken by the general population of the Holy Land at the time that developed initially in Aramaea. It's related to languages like Syriac. The word "rabbouni" for instance is Aramaic, not Hebrew, so we know that's what it means when it says in Greek "in Hebrew." Moreover, a Greek translation is given immediately after, which we wouldn't expect of an audience that knew what "rabbouni" meant—but that speaks more to whom the NT was addressed, and I'm not an expert in the redaction history of John.
Edit: Adding this in for emphasis. There are Aramaic translations of the Hebrew Bible that were very popular in the Second Temple period when Jesus lived. They're called Targums (or Targumim if you want to be technical) and demonstrate that the local population was using Aramaic as a common scriptural language alongside/instead of Hebrew. The books of Daniel and Ezra were also written in Aramaic and those were written in probably the 7th-6th centuries BC.