Are there forces which do not involve a change in momentum? I am familiar with the equation $$\vec{F}=m \vec{a}$$
I am wondering as to whether it is possible for something to exert a force on another object without changing the momentum of said object. My question is: is force defined as a change of momentum? If that is the case, my question is pointless. But if it is not. What is the underlying framework? 
Also, when some phenomena causes the spins or angular momentum of subatomic particles to change, does it mean that the phenomena caused the particle to accelerate or decelerate? What is force in the framework of quantum mechanics? Does the definition of force only apply in classical mechanics (Sidenote: I am not very familiar with quantum mechanics. I apologize beforehand...)
 A: No, all forces involve a change in momentum.
In classical mechanics force is defined as a change in momentum.
In quantum field theory particles interact via exchanging one or more bosons (see feyman diagrams). These bosons always have momentum and therefore the momentum of the interacting particles changes as well.
A: You can have multiple forces exerted on an object that add to zero.  Then there will be no momentum change.  Think of the two of us leaning against opposite sides of a door with the same force.  The door does not change momentum, nor does either of us.  I am exerting a force on my chair as I sit here.
A: Actually Newton's second law is better stated as $$F=\frac{dp}{dt}$$ and this is even valid in relativity, both SR and GR, expressed in the right way
$$
f^\mu = \frac{dp^\mu}{d\tau}=m\frac{du^\mu}{d\tau} = m u^\nu\nabla_\nu u^\mu
$$
(for massive particles)
so classically forces are always imply a change in momentum.
In QFT the classical concept of force is not useful and we talk about interactions but generally they also change momentum.
More directly answering your question, I believe a force is not defined as the change of the momentum but as the cause of that change. 
A: A universal relation is that the force exerted on an object equals the time derivative of momentum. No force, no momentum change, vice versa.
A: Not exactly what you were asking for, but any force exerted perpendicular to the direction of motion does not change the magnitude of momentum -- though it does change the direction.
Two examples are the force exerted by a uniform magnetic field on a moving charged particle, and the force of gravity on a satellite in a perfectly circular orbit.
But momentum is inherently a vector quantity, so this does not really answer your question.
A: As I interpret your question, there is one (and only) situation where "forces do not involve a change in momentum" and that is when the vector sum of the forces is zero.  However, if you meant "net force," then the answer is no.
