What will happen to the Trinity when the Bride of Christ is revealed? Are there any Christian teaching that address What will happen to the Trinity when the “Bride of Christ” is established, will it then be a Quadrinity? 

Revelation 21:9 And there came unto me one of the seven angels which had the seven vials full of the seven last plagues, and talked with me, saying, Come hither, I will shew thee the bride, the Lamb's wife

Revelation 22:17 And the Spirit and the bride say, Come. And let him that heareth say, Come. And let him that is athirst come. And whosoever will, let him take the water of life freely.
No, at least in the context of mainstream Nicene (including Protestant, Catholic and Orthodox) I don't think there is any teaching about this. The reason being simple: it is a non issue. The very idea of this is not based on a classic understanding of the Trinity at all and therefore is not a problem for normal Trinatarian theology.

The basic idea here is that God is ONE being. Never mind the person aspect of his nature for the moment, that is irrelevant to the difficulty you are posing. You have one being, then a separate entity and a relation between them. The one being (esp when that being is God!) doesn't fundamentally change in nature based on the state of relationships with other beings. It is what it is.

Now you can throw the three persons of the Trinity back in the equation, but nothing changes. They are still God and God still has a relationship with his chosen people, his bride.