If gravity - a field force - has an elementary particle, the graviton, why don't other field forces like magnetic fields have their elementary particles? I mean, why isn't there a magneton? Or, what elementary particle is associated with the magnetic field? Is there a boson for the magnetic field?
If one considers the magnetic field to be a special type of EM field with 0 amplitude electric field, then should you expect to detect a photon when you place a photon detector near a magnetic field?
The gauge boson associated with the magnetic field is the photon.
Electric and magnetic fields are in effect different views of the same thing, i.e. the electromagnetic field, and the gauge boson for the electromagnetic field is of course the photon.
Consider you are looking a static charge, which obviously has just a static electric field. But now suppose I am moving relative to that charge. This means the charge is moving relative to me, and a moving charge generates a magnetic field. So you see an electric field generated by the charge while I see a magnetic field. That's why I say electric and magnetic fields are just different views of the same thing.
Footnote: I see Lupus Liber has added an answer that goes into more detail about how the electric and magnetic fields are different views of the EM field, and I recommend reading his answer though you may find it hard going. You might also be interested to read the answers to Do photons truly exist in a physical sense or are they just a useful concept like $i = \sqrt{-1}$?.
There is a particle mediating the electromagnetic interaction: the photon. In the quantum version of electromagnetism (which is a particular example of a quantum field theory), the existence of mediator boson particles for forces is implied.
The following may be worth mentioning:
We say "electromagnetic" (and not "electric" or "magnetic") because this is the only meaningful Lorentz invariant label for this interaction: Magnetic and electric fields are only blocks in the electromagnetic field strength four-tensor, and therefore, are "rotated" into each other under Lorentz transformations. This means that statements like "${\bf E}\neq0$ and ${\bf B}=0$" are true only in a particular frame of reference. This is a classical physics statement, which holds even before quantizing.
By "photon" we mean a quantized plane wave mode of the field. Such a mode has a definite four-momentum. Note, however, that it does not correspond to the case of a classical field configuration. Such a configuration can be constructed by superposition of plane waves and of different multiplicities of modes, which corresponds in the quantum theory to states with various number of photons and various momenta.
It is useful to redefine what a photon is by subtracting the expectation value of the field in the ground state (which is called "vacuum"), given the boundary conditions. This "vacuum expectation value" (VEV) part will automatically obey the classical (in the electromagnetism case: Maxwell's) equations of motion, and the newly defined photons are the quantized fluctuations on top of the VEV.
Detecting photons: yes, this is indeed implied, though practically difficult to measure. For example, a $e^+e^-$ pair may be produced. The pair may emit photons. Since each such emission has a low probability, due to the weakness of the electromagnetic interaction ($\alpha\approx 1/137$), such quantum effects occur with very small probabilities, but they are certainly predicted by quantum electrodynamics. I don't know about the experimental status of this.
About 40 years ago there was an intensive search for the magnetic monopole or magneton. If it were found, the theory of the electromagnetic field would become substantially more complex. However, the magnetic monopole was never observed and the theory remained unchanged, as described in the answer of John Rennie.
This article explains the details:
https://en.wikipedia.org/wiki/Magnetic_monopole
Nevertheless, there have been some claims of the discovery of the magnetic monopole. These results have not been reproduced by others and therefore not accepted by the scientific consensus:
Today, the name of "magneton" is used to describe physical constants of magnetic moment along with other concepts thus creating ambiguity:
why don't other field forces like magnetic fields have their elementary particles?
The source of electric fields are electric charges. So the subatomic particles electron and proton are sources of electric fields. To observe an electric field negative and positive charge(s) have to be separated.
Somehow the same one can say about magnetic fields. The subatomic particles obey a magnetic dipole moment. This is an intrinsic (existing independent from some circumstances) property of these particles. To observe an magnetic field the involved particles have to be aligned with their magnetic dipole moments.
If one considers the magnetic field to be a special type of EM field with 0 amplitude electric field, then should you expect to detect a photon when you place a photon detector near a magnetic field?
In the case an electron approaches to a nucleus we indeed can observe the emission of photons. Another example for the emission of photons is the acceleration of electrons, best observable in an antenna rod where electrons get accelerated back and forth the rod. The interesting fact is that if one observe this radiation from an antenna the radiation is composed of an electric and a magnetic field. So an electron - with its electric charge and its magnetic dipole moment - emits EM radiation. But you would not be able to detect a photon near an electric nor near a magnetic field.
To describe the interaction of electric fields among themselves, magnetic fields among themselves and moving charges in magnetic fields one uses the construct of virtual photons. How this interaction happens in a closer view isn't in the focus of today's physic.
