Answer
Yes. No.
Work Step by Step
A particle moving at constant velocity has no net force on it.
If the magnetic field $\vec{B}$ is zero, and the charge moves at constant velocity, we can conclude that the electric field $\vec{E}$ is zero because the electric field always exerts a force on a charged particle, $q\vec{E}$. In other words, an electric field always exerts a force on a charged particle, so the absence of an electric force always means there is no electric field.
However, if the electric field $\vec{E}$ is zero, and the charge moves at constant velocity, we may NOT conclude that the magnetic field $\vec{B}$ is also zero. If the particle’s velocity $\vec{v}$ is in the same direction as the external magnetic field $\vec{B}$, or in the opposite direction, then the force on it is zero. In other words, a magnetic field does not always exert a force on a charged particle, so the absence of a magnetic force does not necessarily mean there is no magnetic field.
