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Yes, there is school of theological nihilism influenced by Nietzsche, known as the "Death of God" movement. Probably the most famous individual associated with it is Paul Tillich. This work explores the idea that the traditional concept of God has "died", and the extent to which faith (or reason, or ethics, etc.) can still exist in such a world. We can argue about whether this sort of thing is nihilist, or existentialist, or both, or not really either; but in any case it comes out of that general area of thought.
It should be distinguished from the older and more "respectable" pursuit of apophatic theology. This is an approach to God founded on describing God in "negative" terms (he is not evil, he is not limited, etc.). Typically, this theology is mystical rather than rational, and Eastern rather than Western. A standard idea in apophatic theology is that God does not exist, meaning that the mode of his "existence" transcends the way in which ordinary everyday stuff is said to "exist". Contemplation of paradoxes associated with the divine is also part of apophatic practice - without necessarily trying to solve them in the manner of rationalist theology. Apophatic theology has influenced nihilistic theology but they are not identical.
Another relevant writer is Kierkegaard for his exploration of paradox and despair in a Christian context. He is not a nihilist but there is a good deal of later work which responds to him. I would suggest Fear and Trembling as a good starting point (because out of the two books I've read by him, it was the one I liked the best). In it, he works through the story of the Binding of Isaac with an emphasis on the supposed moral of the story, Abraham's greatness through faith; this is contrasted with the anxiety and subjectivity of Abraham's personal experience, in the context of an apparently immoral command to murder his son.
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Negative theology, for example as it was expounded by Pseudo-Dionysius Areopagita and several medieval Christian mystics such as St. John of the Cross or Meister Eckhart is explicitly nihilist in its content. This theology explains God as a form of Emptiness, as something completely unkowable to the intellect. Therefore negative theology can be interpreted as "nihilist" or "proto-nihilist".
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Your question is not an easy one to answer in that nihilism takes many shapes and forms, and therefore it is hard to pin down how nihilism relates to any particular part of the Bible.
For instance the commonly known divisions of nihilism fall into the following categories Metaphysical nihilism, Epistemological nihilism, Mereological nihilism, Existential nihilism, Moral nihilism & Political nihilism
Most Christians that I know would subscribe to Metaphysical nihilism, prior to Genesis 1:1 in that there were no concrete (material) objects.
To embrace Epistemological nihilism would be to dispute that there was a tree of the KNOWLEDGE of good and evil in the garden of Eden.
Mereological nihilism would be denying creation.
Existential nihilism which you seem to be referring you question citing Solomon is tantamount to ascribing Creation to be a meaningless effort.
Moral nihilism would dictate that the 10 commandments are only a fabrication with no intrinsic meaning.
And if Law which is denied as part of Political nihilism has no meaning and really does not exist then God never gave the Law to Moses on the mount. (check out Leo Strauss)
You might also care to read Martin Heidegger's interpretation of Nietzsche.
For my part other than the idea that before creation nothing material existed, none of these other ideas could be accepted by Christians, I most assuredly cannot!
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Check out Gianni Vattimo's work. He's a modern Italian philosopher who identifies as both a nihilist and a Catholic. His main interests are Heidegger, Nietzsche and Rene Girard.
To him, the death of God represents the end of metaphysics (which he deduced is a form of violence), as revealed by the death of Christ. After Christ's death, we are freed from the law (metaphysics) and are free to focus on communities which work toward common goals (a process synonymous with the Holy spirit). The New Testamant destruction of the law is considered the ultimate nihilistic act.
Mine is a drastic paraphrasing but he's very interesting.