Do we have viscous force acting between two layers Frictional force between solids operates even when they do not move with respect to each other. Do we have viscous  force acting between two layers even if there is no relative motion?
 A: 
Do we have viscous force acting between two layers even if there is no relative motion?

No.  From the Wikipedia article on viscosity:

In general, in any flow, layers move at different velocities and the fluid's viscosity arises from the shear stress between the layers that ultimately opposes any applied force

When the fluid is stationary, there's no velocity gradient, hence no shear stress.
A: 
Frictional force between solids operates even when they do not move
  with respect to each other.

There's a little confusion here. Force doesn't operate. It's either applied or not.
Suppose my car, which weighs about 2000 lb, has its brakes locked, so it can't roll.
Suppose the friction coefficient of the rubber tires on the pavement is about 0.5.
What does that mean?
That means if I want to shove it forward I need to push it forward with more than 1000 lb force, to overcome friction.
If I only push with 500 lb force, then only 500 lb force is applied (and it won't move).
If I push with 0 lb force (i.e. not push), then 0 lb force is applied (and of course it won't move).
So when solids do not move with respect to each other, it does not mean that frictional force is operating.
That might help you to understand viscosity a little better.
A: The fundamental reason why solids and fluids are different in this respect is broken symmetry. The fluid doesn't break translational invariance, so that there is a continuous notion of conserved momentum in the fluid, which can vary arbitrarily slowly from point to point. In a solid, the translation invariance is broken to a discrete subgroup, so you need to move the entire solid with a given velocity to give it momentum, there are no statistical states where part of the solid is moving with one speed and another part is moving at a slightly different speed (since this will break the solid lattice).
The result is that when you have two solids on top of each other, either they are both acting as one big solid (the static friction case), or else one is sliding on top of the other with a boundary between the two velocities which is only a few atomic spacings wide. This is the fundamental difference between solid and fluid.
A: As dbaseman mentions in fluids the viscous forces are present because of velocity gradient. At molecular level in fluids viscous forces are present due to transfer of momentum between two layers as there is exchange of molecules (with unequal velocity) between the different layers. This happens when there is relative motion between the two layers. Therefore when the fluid is at rest there is no relative motion and hence no net transfer of momentum and therefore no viscous forces.
In case of solids the friction forces are present due to intermolecular forces between the surfaces of two solids. This is does not require any transfer of momentum and hence are observed even when the solids are at rest relative to each other. You can think of static friction as the force required to break intermolecular bonds at the surface. If you apply less than required force, the bonds will not break and hence the observed resistance.
Note that the friction and viscous forces are different but they have same effect i.e. to resist any relative motion and, in presence of any motion, dissipate kinetic energy by converting it to thermal energy (heat). So it is often convenient to think of them as the same thing.
