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Answer:

Yes, the three persons of the trinity always cooperate with each other at all times in perfect unity. For example, at the baptism of Jesus, the Son was baptised to fulfill all righteousness, the Father spoke from heaven, “You are my beloved Son; with whom I am well pleased”, and the Holy Spirit descended upon Jesus like a dove. (Luke 3:22) Likewise, at the creation of the world, we see God speaking the world into being, the Word becoming flesh in Jesus (Gen 1:3; John 1:1-3,14), and the Holy Spirit hovering over the waters (Gen 1:2). The Triune Godhead all have the same attributes and cooperate in the sense that they have one common purpose. They cooperate in all that they do.

Where all creation and time is concerned Ephesians 3:11 tells us that “This was according to the eternal purpose that he has realized in Christ Jesus our Lord”.

The word purpose here (Greek: prothesin) can be translated as “plan” or “resolve” and is in the singular. God has one overarching purpose or plan that is affirmed by the Triune Godhead.¹ And this purpose is eternally realized (made a reality) in Jesus. Here we see that the Triune Godhead cooperates in all that they will to do, and this purpose is an eternal or everlasting one without a beginning or end. The plan and purpose of God has been already been established from eternity past, and they all work in cooperation towards that end.  In other words, the Triune God does not come together every now and then to decide what decisions they need to make. From all eternity past, they have one single overarching purpose that will be carried out in perfect unison.

The Trinity cooperates with each other in all that they do, and will to do.

[1] Robert Reymond, A New Systematic Theology of the Christian Faith, second ed., p.463