The New International Version (NIV) is based upon “the Koine Greek language editions of the United Bible Societies and of Nestle-Aland.”1 Specifically, it is based on the Nestle-Aland 27th edition.
The Greek text of the NA27 states:
Ϛʹ ὅτι δέ ἐστε υἱοί ἐξαπέστειλεν ὁ θεὸς τὸ πνεῦμα τοῦ υἱοῦ αὐτοῦ εἰς τὰς καρδίας ἡμῶν κρᾶζον αββα ὁ πατήρ NA27
On the other hand, the New King James Version, like the King James Version before it, is based upon the Textus Receptus, which states,
Ϛʹ Ὅτι δέ ἐστε υἱοί ἐξαπέστειλεν ὁ θεὸς τὸ πνεῦμα τοῦ υἱοῦ αὐτοῦ εἰς τὰς καρδίας ὑμῶν, κρᾶζον Αββα ὁ πατήρ TR, 1550
The difference in translations is caused by different pronouns, ἡμῶν (“our”) in the NA27 and ὑμῶν (“your”) in the Textus Receptus. Contextually, ὑμῶν (“your”) seems to be more sensible; it is simply nonsensical for the apostle Paul to say, “Since you are sons, God has sent forth the spirit of His Son into our hearts.” If ἡμῶν (“our”) was the legitimate reading, we might expect the Greek to have had the following:
Ϛʹ Ὅτι δέ ἐσμεν υἱοί ἐξαπέστειλεν ὁ θεὸς τὸ πνεῦμα τοῦ υἱοῦ αὐτοῦ εἰς τὰς καρδίας ἡμῶν, κρᾶζον Αββα ὁ πατήρ
6 Since we are sons, God sent forth the Spirit of His Son into our hearts, crying, “Abba, Father.”
Both variants (ἡμῶν and ὑμῶν) are supported by a plethora of witnesses. Therefore, it seems we must rely on context to determine the true reading (this, of course, is my opinion; I’m not a textual critic or scholar).
Here are the witnesses according to Constantin Tischendor’s critical apparatus:
Barker, Kenneth L. The NIV: The Making of a Contemporary Translation. Colorado Springs: International Bible Society, 1991.
1 p. 54