Does God have a wife?

God does not have a wife. The institution of marriage only began in Genesis 2, and God predates this creation account. 

GEN. 2:21 So the Lord God caused a deep sleep to fall upon the man, and he slept; then He took one of his ribs and closed up the flesh at that place. 
GEN. 2:22 And the Lord God fashioned into a woman the rib which He had taken from the man, and brought her to the man. 
GEN. 2:23 Then the man said,
“At last this is bone of my bones,
And flesh of my flesh;
She shall be called ‘woman,’
Because she was taken out of man.”
GEN. 22:24 For this reason a man shall leave his father and his mother, and be joined to his wife; and they shall become one flesh. 
GEN. 22:25 And the man and his wife were both naked, but they were not ashamed.

Moreover, if we look to the scriptures to why the marriage institution was initiated, we will find this:

GEN. 2:18 Then the Lord God said, “It is not good for the man to be alone; I will make him a helper suitable for him.” 

God has never been alone due to the divine Trinity (God the Father, God the Son, and God the Spirit). The covenant of marriage was introduced and intended for man not to be alone once sin entered the world, therefore, marriage is not an eternal relationship:

MATT. 22:29 But Jesus answered and said to them, “You are mistaken, since you do not understand the Scriptures nor the power of God. 
MATT. 22:30 For in the resurrection they neither marry nor are given in marriage, but are like angels in heaven.  

The marriage relationship is a picture for humanity of something far greater, our relationship with Christ. Once we enter Heaven there is no longer a need for a picture because we will be in the presence of Christ forevermore.