Does a friction force act on a body on a flat surface with no external force? I know that if an object is on an incline, there is friction as a reaction to the component of the object's mass that is parallel to the surface.
I know that if an object is on a flat surface and an external, horizontal force is applied, there is a friction force as a reaction.
But what about an object sitting on a flat surface with no external force? Does the friction force not "exist" or not apply? If I were to draw a FBD, would it just be the gravitational force and normal force reaction?
 A: Friction is a self-adjusting force that opposes relative contact motion between two surfaces. If a surface is not perfectly smooth, friction will always be there. If a block is on a flat surface, friction adjusts itself to zero as there is no tendency for the block to rub against the flat surface. If you were to apply even the slightest force in the horizontal direction, the frictional force would try to counteract that.
A: 
I know that if an object is on an incline, there is friction as a
  reaction to the component of the object's mass that is parallel to the
  surface.

It is more accurate to say that there is static friction acting up the plane that is equal and opposite to the component of the gravitational force acting on the object down the plane. Static friction prevents motion of the object down the plane. But if the component of the gravitational force acting down the plane exceeds the maximum possible static friction force, the object will slide down the plane. Then the friction force acting up the plane becomes the kinetic friction force. If you do a free body diagram as @Eagle suggested, you will see this.

I know that if an object is on a flat surface and an external,
  horizontal force is applied, there is a friction force as a reaction.

That is correct.

But what about an object sitting on a flat surface with no external
  force? Does the friction force not "exist" or not apply? If I were to
  draw a FBD, would it just be the gravitational force and normal force
  reaction?

A friction force only exists in opposition to an external applied force. It is zero if the applied force is zero. As an applied external force increases, the static friction force increases by the same amount so that the net force is zero and there is no relative motion between the object and the surface. This continues until the maximum static friction force is reached, which is for a flat surface $f_{max}=μmg$ where $μ$ is the coefficient of static friction. Then the object slips and kinetic friction takes over.
Hope this helps.
A: Consider a body on a surface. What we call the friction between the body and the surface is actually the averaged effect of interactions between countless atoms where the body and surface meet. When the body is stationary on the surface, all of those individual interactions between atoms cancel each other out- they are in equilibrium, and the net force is zero. So, overall there is no net frictional force acting on the body- if that wasn't the case, friction would cause the body to move. 
When a small external force is applied to the body it will shift fractionally and all the millions of tiny forces will change somewhat, some increasing and some relaxing until equilibrium is reached again. If the force is gradually increased the body will continue to move fractionally until the point at which equilibrium can no longer be maintained- beyond that point further increases in the external force will accelerate the body across the surface.
A: If maximum static friction force would be expressed in vector form it would be:
$$ \large{\vec{F}_{max} = -\mu_{s}\cdot\|\vec{F}_n\| \cdot \hat{\textbf{u}}_{\perp F_n}} $$
here,


*

*$\mu_{s}$ - static friction coefficient

*$\vec{F}_n$ - normal force

*$\hat{\textbf{u}}_{\perp F_n}$ - unit vector of applied external force perpendicular to normal force vector


So, there is no static friction force if :


*

*surfaces are friction-less 

*no normal force is exerted on body (for example, weightlessness mode in free-falling bodies)

*no external force applied perpendicular to normal force


As a side-note,- if static friction force would exist without external force applied,- then body which have been put on flat surface, would have started to move itself with acceleration (second Newton law), without any apparent reason. Thus it would break causality.
However, surface features which causes friction force,- called asperities-, exists always no matter friction force is induced or not. Asperities:

As a bonus points. Typical earthquake is caused by the same friction mechanism when static friction force is not able to hold-on pair of drifting tectonic plates glued together anymore.
