No, we will not see the Trinity in heaven. The Trinity is the biblical teaching that God exists in three distinct and simultaneous persons: the Father, the Son, and the Holy Spirit. There are not three gods, but only one. We will not be able to see the totality of what the Trinity. The Bible tells us that God the Father cannot be seen, ever.
1 Timothy 6:13–16, "I charge you in the presence of God, who gives life to all things, and of Christ Jesus, who testified the good confession before Pontius Pilate, 14 that you keep the commandment without stain or reproach until the appearing of our Lord Jesus Christ, 15 which He will bring about at the proper time—He who is the blessed and only Sovereign, the King of kings and Lord of lords, 16 who alone possesses immortality and dwells in unapproachable light, whom no man has seen or can see. To Him be honor and eternal dominion! Amen."
Verse 16 is a crucial verse. I supplied the context so you could see that both the Father and the Son are being discussed. You will notice in verse 16 that it says, "whom no man has seen or can see." Since Jesus has obviously been seen by the people he walked the earth with, we can conclude that this is not speaking of Jesus. Therefore, in the context since it is speaking of God, in this case, the Father, then it must be the Father who has never been seen nor can be seen. We conclude that the full manifestation of the Trinity will not be seen in heaven because the Bible says the Father cannot be seen in verse 16.
However, some people say that God the Father was seen in the Old Testament in such verses as Genesis 17:1; 18:1; Exodus 6:2-3; 24:9-11; Numbers 12:6-8, etc. But we can clearly and unequivocally state that it was never the Father who was seen because of 1 Timothy 6:16 as well as another verse uttered by Jesus.
John 6:46, "Not that anyone has seen the Father, except the One who is from God; He has seen the Father."
Jesus says that no one has seen the Father except the One who is from God. Of course, the only one who has seen the Father is Jesus himself since he is the second person of the Trinity. So, it is consistent with Scripture to say that God the Father has never been seen in the Old Testament. They were seeing the pre-incarnate Jesus. So, to reiterate, we will not see the Trinity in heaven. But, who then will we see?
Jesus is the representation of God
So, who then will we see in heaven if not the Father? I think it is safe to say that we will see Jesus who is the representation of God the Father.
Hebrews 1:3, "And He is the radiance of His glory and the exact representation of His nature, and upholds all things by the word of His power. When He had made purification of sins, He sat down at the right hand of the Majesty on high."
Another verse that is important here is found in 1 Corinthians.
1 Corinthians 1:9, "God is faithful, through whom you were called into fellowship with His Son, Jesus Christ our Lord."
What I want to point out, is that we are directed by God (the Father) to have fellowship with Jesus. This is a necessity because God is so inexpressibly majestic, holy, pure, and infinite that there is no way we as mere biologically limited creatures can dwell in his presence. He is too vast for us to comprehend. Therefore, we are given the privilege of experiencing the presence of God through the person and work of Jesus. So, in heaven is safe to say that we will be experiencing the presence of God the Father through our relationship with God the Son.
Will we see the Holy Spirit in heaven?
The Bible only tells us that the Father cannot be seen. This is not said of the Holy Spirit. Furthermore, we know in the New Testament the Holy Spirit appears at the baptism of Jesus (Matthew 3:16) in what appears to be "tongues as of fire" (Acts 2:3). So, there is a kind of manifestation that is perceivable by people regarding the Holy Spirit. So, it is possible in heaven that we will see the Holy Spirit in some manifestation. However, I want to clarify that this is conjecture. The Bible does not explicitly state we will be able to see the Holy Spirit in heaven, but since he is seen on earth, it is certainly possible we will see him in heaven.