Are there forces which do not involve a change in momentum?

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.

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.

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.