Is a charged particle in motion an electric current? Do a moving charge let's say electron linearly with constant velocity constitue an electric current? If yes what would be it's magnitude? 
 A: The most "authoritative" answer on this is to look at Maxwell's equations, because they are at the heart of EM theory.
And the answer there is yes, it does count as "current" for their purposes, in that it contributes to the $\mathbf{J}$ term. The $\mathbf{J}$ part is defined as
$$\mathbf{J} = \rho_q \cdot \mathbf{v}$$
where $\rho_q$ is the charge density field and $\mathbf{v}$ the velocity field. A particle corresponds to a delta spike in $\rho_q$, with a point assignment of velocity (not a delta) in $\mathbf{v}$ at its position, and so as long as it is moving, then $\mathbf{J}$ also has a delta spike. The more familiar scalar current $I$ is a related quantity that is the flux of current density through a surface, i.e.
$$I = \int_S \mathbf{J} \cdot d\mathbf{A}$$
In this case, the charged particle also may provide such a current, but it will only do so when it is exactly coincident with $S$. If we consider $S$ a general surface in its path, that means that $I$ will then be a delta spike in time, i.e. an "impulse" of current, at exactly the point where the particle crosses the surface.
Now, you may wonder why that for "real" currents, which are still comprised of the passage of many individual electrons through a surface, we don't just take them as a high-frequency set of delta spikes. The answers are at least three:
One is that it the classical model of the charge in an extended material like a piece of wire is not atomic; it is a continuous distribution. Thus a point particle is considered only as a limiting case. This is the case for classical physics in general, including for mechanics, i.e. the motion of matter. Rigid bodies and fluids are also treated the same way. Classical matter is continuous; it does not feature "atoms" and that includes particles like electrons. Keep in mind that science in general is about creating models, and it does not give us "absolute truths". There is nothing wrong with modelling matter with a continuous distribution so long as you don't look too closely at it.
The second reason is that, if you do decide to look close enough, the classical model of a bunch of points in motion doesn't work: quantum mechanics takes over at that point, and so to do electrodynamics, you will have to use quantum electrodynamics (QED).
The third, and final, reason is that while we can indeed make a classical particulate model within the Maxwell framework, i.e. treat $\rho_q$ as a huge swarm of delta spikes, it is not convenient to do so. In light of that the classical continuous model works for macroscopic situations, and that the microscopic cannot be treated faithfully with classical mechanics, it becomes simply an unnecessary complication.
A: Loosely, yes, it's a flow of charge. 
Strictly no. Current (in a quantifiable sense) means charge flowing through a given area per unit time. I don't see how you can apply this definition to the one-off passing of a charge through an area, even if you know the charge's velocity. Assigning a the time taken would be arbitrary.
But if you know that there are n particles per unit volume, each of charge q and moving with velocity v at right angles to an area A, then the current through A will be$$I=nAve.$$ 
A: If you are asking about a single electron moving, then no, it does not constitute a current.
When we talk about a current, say through a wire, we are looking at a net movement of charge through some defined surface over some time interval. So it does not make sense to define a current for just a single charge. If you were to try to do this, you would count one electron, and then your "magnitude" of this current would completely depend on the time interval you are looking at, going to $0$ the larger the time is after this charge passes through this surface.
If you are thinking about looking at currents in magnetostatics, then you need currents that are constant over space and time. This is analogous to a single charge being the "unit" of electrostatics. A small current element is the "unit" of magnetostatics (compare Coulomb's Law to the Biot-Savart Law). 
Even if the currents are not constant in space or time, we still need a net flow of multiple charges in order to discuss currents. The idea of current is more of an "on average" thing. A single electron moving in a line is just that, a single electron moving.
A: How do you define flow of water ? You may say it's the amount of water that pass through a cross section in unit time. But what can you say about the one water molecule ? The same argument goes for an electron. 
We defined electric current for practical purpose - it's not as 'fundamental' as an electron in physics. If you want to define the electric current for single electron, it's not wrong with it, but it's might or might not useful (like the flow of 'single' water molecule).
A: I've been trying to understand this as well.  This is what I've been able to figure out so far. I welcome comments and corrections.
A charged particle in motion is electrically equivalent to an electric current even if you can't calculate the current in the conventional sense of measuring Coulombs per second.
We can say this because a current made up of many moving elections in a wire will have a corresponding magnetic field, but a single moving electron will also have a corresponding magnetic field even if trying to compute the current as Coulombs per second is illogical and not possible.
This was the problem noted by Maxwell in the mid-1800s and addressed by his modification of Ampère’s law that defines the relation of the current to the magnetic field.  It required the addition of a term that computed the effective total current by adding a term for changing electric field strength.
Ampère’s law relates a current through any closed area to the integral of the magnetic field around the border.  A current through the area defines the strength of the magnetic field at the border, but an electron approaching the area, which creates a growing E field through the same area, has the same effect on the magnetic field as a "real" current through the area.
This web page has a very good discussion of this issue though does not talk directly about the subject of this question:
https://opentextbc.ca/universityphysicsv2openstax/chapter/maxwells-equations-and-electromagnetic-waves/
So from the above web page, we see the equivalent current created by the changing Electic field in an open surface is:
ₒ dE/dt
Where ₒ is the Vacuum permittivity of space.
The value of this term will change as the electron approaches, goes through, and leaves, the surface used in the equation. So there is not a single current "value" for the electron moving through space but it acts as a changing equivalent current in Maxwell's equations.
A: For a particle with charge $q,$  velocity $v$, and position $x'$ you can define the current:
$$I(x)={q}{v}\, \delta( x-x')$$
where $\delta$ is the Dirac-delta. Note that this is not a stationary current, so we can't apply formulas from magnetostatics.
A: The motion of a single charged particle all by itself does not constitute a current. The notion of a current implies that there is a flow of charge bearing particles, whether this be electrons or protons.  Notably, the notion of a current arose phenomenologically well before the atomic notion of electron was postulated and discovered.
