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Are all fields in the universe we know of quantum fields? Do all fields that exist must be inherently quantum in nature? How about fields that are yet to be discovered ie.
Currently all fundamental fields are quantum, except for gravity. For this reason Quantum Gravity is a hot area of research, but the full Quantum Gravity theory has not been developed yet. Why not? The challenge is not just technical, but conceptual. On one hand, the Quantum Field Theory cannot consistently co-exist with any classical theory.
If the Quantum Field Theory is correct, then gravity must be quantum.
 A New Way of Visualizing Quantum Fields
On the other hand, gravity cannot be just another quantum field theory, because gravity bends the space and time "the background" , on which the Quantum Field Theory is based, and this creates unreasonable challenges time is steady and independent in QFT, but depends on the field and is dynamic in GR that technically result in non-renormalizability of quantization. The only logical way to resolve this contradiction is to admit that both theories, General Relativity and Quantum Field Theory, are approximations of another unknown yet theory that in itself is neither General Relativity nor Quantum Field Theory.
As mentioned in the comment of Prahar above, chances are that gravity will be quantized in some way, but Quantum Gravity will not be a standard Quantum Field Theory. Other possibilities also have not been ruled out, such as that gravity may not have a quantum nature or have a nature that would change our understanding of "quantum" and what exactly we mean by it. Thus the answer to your question is that no one knows yet.
We believe that fundamentally the universe is quantum mechanical.
quantum field theory
So we would expect all fundamental fields to be quantum in nature. However that does not mean that all fields are quantum.
The laws of physics depend are scale-dependent Joseph Conlon gives a very good exposition on this in Why String Theory? If the scale of the problem lies in the classical domain then the fields of interest will be classical in nature. In principle we can solve such a problem with quantum fields, but in practice it does not quite work that way.
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quantum field theory
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QED , quantum electrodynamics - a relativistic quantum theory of the electromagnetic interactions of photons and electrons and muons. QCD , quantum chromodynamics - a theory of strong interactions between elementary particles including the interaction that binds protons and neutrons in the nucleus ; it assumes that strongly interacting particles hadrons are made of quarks and that gluons bind the quarks together.
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He contends that it is more logical and easier to teach the various sub-fields in the reverse order of their historical development, so he elucidates the physics from quantum field theory to classical mechanics from a set of common basic principles in a unified way. Instead of quantum field theory , we have program- matic and hyper-personalised messaging.
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Particles in our universe are described by relativistic quantum field theory , which combines quantum mechanics with Einstein's theory of relativity. The quantum field theory approach to quantum mechanics is on a solid footing. Beyond quantum fields: a classical fields approach to QED. The first volume can serve as a textbook for an introductory graduate course, and covers kinematical and dynamical aspects of classical relativistic field theory, operator methods and functional integral methods for relativistic quantum field theory , non-relativistic quantum mechanics, and quantum field theory at non-zero temperature.