I doubt it, for the exact reason you pointed out: Hooke's law is not a "fundamental" law. It results from the forces exerted between neighbouring atoms. As I expect you already know, there's a stable equilibrium distance two atoms can be situated from one another without exerting a net force on one another. Push them closer and they repel; pull them apart and they attract. It turns out that the force varies roughly in a linear fashion near this equilibrium distance, which explains Hooke's law. The full law describing the inter-atomic forces can be derived using quantum mechanics, which does have a relativistic formulation.