In John 20:17 does Jesus have a God?

Jesus the man did have a God, the same one the rest of us have. This is in no way contradictory to a triune God.

Your confusion apparently stems from separating Jesus the Divine Spirit from Jesus the human man. The timing of her seeing Jesus and his death is in that it was the risen Christ she saw (the human Jesus). That human identity had not ascended into Heaven. His human identity had to ascend into Heaven in order to fulfil:

Psalm 16:9 and 10 KJV  Therefore my heart is glad, and my glory rejoiceth: my flesh also shall rest in hope. For thou wilt not leave my soul in hell; neither wilt thou suffer thine Holy One to see corruption. 

If this be true of King David how much more true is it for the Christ?

At this point Jesus the man was in his ressurrection body. That body is like unto our ressurection body, with one great difference, that being in the fact that Jesus body was reurrected before decay had begun.

Jesus resurrection body is the exact same body that hung on the cross, that is why Thomas could have put his finger into the hole in Jesus side. our resurrection body will will not be the same. The reasoning behind that is that our bodies have been corrupted in sin; while Jesus body was not. If we in our state of salvation, are to be acceptable to enter Heave we must of necessity be completely free of sin. In other words we must be perfect even as God is perfect, and that means sinless.

Matthew 5:48 KJV  Be ye therefore perfect, even as your Father which is in heaven is perfect. 

We cannot know what our resurrection body will be like, but we might keep this in mind.

Mark 12:24 and 25 KJV  And Jesus answering said unto them, Do ye not therefore err, because ye know not the scriptures, neither the power of God? For when they shall rise from the dead, they neither marry, nor are given in marriage; but are as the angels which are in heaven. 

Whatever that is it is fine with me!

Hope this helps.

This is not at odds with the belief in God's being triune ('the Trinity'). In fact, 'the trinity creed,' the Athanasian, says this:

(Notice the part highlighted by me)

Whosoever will be saved, before all things it is necessary that he hold the Catholic Faith. Which Faith except everyone do keep whole and undefiled, without doubt he shall perish everlastingly. And the Catholic Faith is this, that we worship one God in Trinity and Trinity in Unity. Neither confounding the Persons, nor dividing the Substance. For there is one Person of the Father, another of the Son, and another of the Holy Ghost. But the Godhead of the Father, of the Son and of the Holy Ghost is all One, the Glory Equal, the Majesty Co-Eternal. Such as the Father is, such is the Son, and such is the Holy Ghost. The Father Uncreate, the Son Uncreate, and the Holy Ghost Uncreate. The Father Incomprehensible, the Son Incomprehensible, and the Holy Ghost Incomprehensible. The Father Eternal, the Son Eternal, and the Holy Ghost Eternal and yet they are not Three Eternals but One Eternal. As also there are not Three Uncreated, nor Three Incomprehensibles, but One Uncreated, and One Incomprehensible. So likewise the Father is Almighty, the Son Almighty, and the Holy Ghost Almighty. And yet they are not Three Almighties but One Almighty.

So the Father is God, the Son is God, and the Holy Ghost is God. And yet they are not Three Gods, but One God. So likewise the Father is Lord, the Son Lord, and the Holy Ghost Lord. And yet not Three Lords but One Lord. For, like as we are compelled by the Christian verity to acknowledge every Person by Himself to be God and Lord, so are we forbidden by the Catholic Religion to say, there be Three Gods or Three Lords. The Father is made of none, neither created, nor begotten. The Son is of the Father alone; not made, nor created, but begotten. The Holy Ghost is of the Father, and of the Son neither made, nor created, nor begotten, but proceeding.

So there is One Father, not Three Fathers; one Son, not Three Sons; One Holy Ghost, not Three Holy Ghosts. And in this Trinity none is afore or after Other, None is greater or less than Another, but the whole Three Persons are Co-eternal together, and Co-equal. So that in all things, as is aforesaid, the Unity in Trinity, and the Trinity in Unity, is to be worshipped. He therefore that will be saved, must thus think of the Trinity.

Furthermore, it is necessary to everlasting Salvation, that he also believe rightly the Incarnation of our Lord Jesus Christ. For the right Faith is, that we believe and confess, that our Lord Jesus Christ, the Son of God, is God and Man.

God, of the substance of the Father, begotten before the worlds; and Man, of the substance of His mother, born into the world. Perfect God and Perfect Man, of a reasonable Soul and human Flesh subsisting. Equal to the Father as touching His Godhead, and inferior to the Father as touching His Manhood. Who, although He be God and Man, yet He is not two, but One Christ. One, not by conversion of the Godhead into Flesh, but by taking of the Manhood into God. One altogether, not by confusion of substance, but by Unity of Person. For as the reasonable soul and flesh is one Man, so God and Man is one Christ. Who suffered for our salvation, descended into Hell, rose again the third day from the dead. He ascended into Heaven, He sitteth on the right hand of the Father, God Almighty, from whence he shall come to judge the quick and the dead. At whose coming all men shall rise again with their bodies, and shall give account for their own works. And they that have done good shall go into life everlasting, and they that have done evil into everlasting fire. This is the Catholic Faith, which except a man believe faithfully and firmly, he cannot be saved.

A man definitionally has a God. Since Christ was and is a man, truly and really, He too had—and has—a God. But His having a God and our having a God is not equivalent, since His is personally, equal with the Father. Any doing of the will of the Father is not subservience, but pure love and obedience.

Since Christ is one Person with two natures, what is said of either of His natures can be said to belong to the one Person. Not that both natures are confused with one another, but that things pertaining to the two natures pertain to but one Person.

Scripture shows they are not at odds

Some examples come to mind to demonstrate this:

"The Lord of Glory" was crucified: 1 Corinthians 2:8. cf. Psalm 24.

"The First and the Last..was dead..and [lives] forevermore." Revelation 1:17-18. cf. Isaiah 44:6.

"God... ... [died]." John 1:1,14; 12:33/19:33. cf. Zechariah 12:10.

Etc. Both 'He is God' and 'He died' can be said of Jesus, while strictly speaking, whereas the Person died because death happened to Him indeed, only one of His natures, the human, can suffer. Similarly, only one of His natures can 'have a God' (Revelation 3:12) as it were—His human nature, "the man Christ Jesus." (1 Tithomy 2:5)