There may be places where the word is ambiguous. What I am looking for is one example that must be all three.
The short answer is: nowhere. But your question is misguided.
You cannot disprove the nature of God as Trinity by looking at how the NT authors use the word "God" (Θεός or Κύριος) because within a few decades after the revelation that Jesus was the 2nd member of the Trinity (as God the Son), the NT authors were still transitioning their terminology from Θεός and Κύριος ("God") to "Father". The word "God" / "Lord" used to refer to Israel's God in the OT for at least 1,000 years. It's understandable that for the sake of continuity, the NT authors were still using the same word "God" to mean God the Father who sent His Son when the sentence context needed it, while gradually reassigning "Lord" for Jesus, which is yet another sign of Trinitarian understanding driving the terminology change. It was also logical to use new words for Jesus (such as Χριστός) and the Holy Ghost because both were newly revealed, bringing what was vaguely latent in the OT to be more explicit in the NT.
Thus the apostles were still in the very early stages of the Trinitarian formula development although they already understood God as Trinity shown by how they talked about worship, baptism, salvation, and eschatology because the roles of the Father, Son, and Holy Spirit are distinct in each of those areas. Please don't confuse formula development (which has to do with words and definitions, like what your question implies) with apostolic understanding of the nature of God. It's like Newton already understood the laws of mechanics before Einstein refined the formula.
For an example of an apostolic understanding of the Trinity and a sign of the transition, simply look at one of the earliest record of the baptism formula in Matthew 28:19: "In the name of the Father, the Son, and the Holy Spirit", which in Greek is εἰς τὸ ὄνομα τοῦ Πατρὸς καὶ τοῦ Υἱοῦ καὶ τοῦ Ἁγίου Πνεύματος. Where is the word Θεός or Κύριος in that formula? Nowhere! But surely God has to do with baptism, don't you think?
Consider @curiousdannii pointing out 1 Cor 2:10-16. This was an even earlier sign of apostolic understanding of the Trinity, since this letter was commonly dated to early 50s rather than Matthew's early 60s. In this passage, the word "God" in verse 10 clearly shows how St. Paul had the Trinity in his mind, since within the same section he explained the how it was possible for humans to comprehend the mysterious nature of God as Trinity by exploring the relationship between God, the Spirit of God, and the mind of Christ.
Other examples of apostolic understanding of God as Trinity can be seen in @Geremia's answer. I understand that these verses don't answer your question, but it's a good list showing how the apostles already conceived God as Trinity within a mere 20-30 years after Jesus returned to heaven from where He came. Recent research has also validated how NT authors's understanding of God came exclusively from Jewish perspective of God's fulfilling His promises rather than too much influenced by Hellenistic ideas (as Philo was). (Example: see this paper comparing John and Philo). (As an aside, the Jewish philosopher Philo was also marginalized by their own people within the Rabbinic Tradition). Therefore we can safely say that the apostolic Trinitarian understanding of God wasn't due to some pagan influence as your other question tried to establish.
Explaining the nature of the one God is the whole point of the doctrine of the trinity. That is its goal and it does so by IDENTIFYING God in MANY different contexts that reveal His UNIQUE attributes usually manifested by His UNIQUE actions.
The Bible identifies God by, 1.) His names. 2.) His titles. 3.)His unique attributes. 4.) His unique actions. 5.) His worship. I am not aware of any other literary, contextual means by which the Bible clearly identifies God. I could be wrong, but I think this list is comprehensive.
If you will examine the Bible thoroughly you should be able to quickly discover that there are three and ONLY three "persons" who are identified as God by the COMBINATION of the literary means listed above.
These persons are each variously... 1.) CALLED by the NAMES of God, (YHWH, and its variants) either directly or indirectly, usually both.
2.) RECOGNIZED with the TITLES of God, (Lord, king, savior, redeemer, etc.
3.) ATTRIBUTED with the UNIQUE characteristic of God, (omnipresence, omnipotence, omniscience, eternality, etc)
4.) CREDITED with the UNIQUE actions of God, (creation, origin of God's word, salvation of men and/or creation, etc.
5.) WORSHIPED and/or given the honor, reverence and position due to God ALONE etc.
I am NOT saying that the Father, Son and Holy Spirit are all consistently, equally and in every mention identified as God in every place they are represented in the Bible by any combination of these 5. Nor am I saying that each person of the trinity is represented equally by ALL FIVE of these means of identifying God.
I AM saying, 1.) that each person of the trinity receives some COMBINATION of these 5 means of identifying and distinguishing God listed above. 2.) ONLY the three persons of the trinity receive some COMBINATION OF THESE 5.
The Bible does not attribute creation (a UNIQUE act of God alone) to any person other than those to whom it ALSO refers to as "God" by name or title and to whom is also attributes other unique actions or attributes that identify "God."
Looking at it from another point, the Bible does not call anyone "God" (in the proper sense - meaning THE one and only God) to whom it does not attribute the UNIQUE actions and UNIQUE attributes of God.
Looking at this from the negative, the Bible does NOT attribute the UNIQUE CHARACTERISTICS of God, such as omnipresence or eternality, to anyone whom it does NOT call "God" (by name and/or title) in the proper sense.
The Bible never attributes omnipresence or eternality to anyone whom it does not call "God" and /or shows being worshiped. Again, there are only three persons in the whole Bible who meet this criteria for identifying God.
The list of examples could go on but I think you get the point. A true understanding (within human limits) of who God IS is essential to a valid relationship with Himself according to His own purpose. No valid, systematic theology can be drawn by reducing or eliminating scriptural evidence. We must account for ALL of God's revelation to come to a realization of who God is and what He has done and therefore what His very nature is.
God has revealed Himself to us in different ways at different times as the Biblical revelation unfolded over time according to His eternal purpose. It is ONLY accounting for ALL of this revelation that we can know Him as He has revealed Himself.
BUT theology is NOT salvation. It is in Christ that we find God fully and sufficiently manifested to us human beings, AS a human being and for His own glory. And it is only through His Holy Spirit that we can know Him and be saved by His grace.
As I've read some of the threads and post around here, some of which ask about, "What is the Trinity." "Can it be explained by "ice water and steam?" Is the four leaf clover like the Trinity etc. Examples are fine and have their place to some. But the Bible itself is better and what I posted is not hard to understand.
You just have to "THINK" when your reading and pay attention to the "CONTEXT." Most of the time the context will answer your questions, even if you have to read the whole chapter, the chapter before and the chapter after to get insight of what's going on.
I was reading a thread about how the Trinity is "rooted" in paganism. It was copied, I don't know by Egyptians way before Jesus Christ appeared on the seen. And then there is the three headed snake and all the rest of these bogus claims. Does anybody really think about how God is clearly identified as three persons IN THE BIBLE is rooted in paganism?
God is Trinity. Therefore all references to God should be taken as referring to the Trinity as a whole unless they are clearly referring to only one of the Trinity.
So almost all references to God in the Old Testament should be taken as meaning the Trinity, except explicit references to the Spirit or Messiah or similar. In the New Testament the exceptions would be explicit references to the Spirit or Jesus or the Father. Note that Jesus can refer to God doing something which may mean the Trinity (which includes himself), in the same way as a committee member might refer to "The committee" (which he is part of) doing something without feeling the need to mention that he is part of it.
However be aware that drawing a sharp distinction between the Persons of the Trinity is usually a bad idea. As a general rule if The Spirit (or Jesus or The Father) does something then God does that thing.
NOTE: In this answer I address how we should interpret the word God with regard to the Trinity when we read it in the Bible. I do not consider how the writers might have understood the issue, and on that I defer to @GratefulDisciple's excellent answer.
For there are three that bear record in heaven, the Father, the Word, and the Holy Ghost: and these three are one. And there are three that bear witness in earth, the Spirit, and the water, and the blood: and these three agree in one," (1 John 5:7-8, KJV).
For many years this was the standard response to a question like yours.