Question

Can you associate a de braglie wave with a particle at rest?

Answer 1

Of course, you can, but will it make sense or not is a more important question. As we remember, the equation is

$$\lambda=\frac{h}{mv}.$$

For a particle at rest, the velocity is zero, and the de Brogile wavelength becomes infinity (the wave becomes a straight line roughly speaking). This does not make any sense.