As long as you correctly exported your atlases from Spine at your desired resolution, this is mostly a texture settings thing on the Unity side.
Try to take a look at your Texture asset settings in Unity, and change the settings to suit your compression, sharpness and filter requirements.
For example, you might want to enable, or disable mipmapping.
There are also texture settings that Unity doesn't expose in the Texture importer inspector, like Texture.mipMapBias
This has to be set through code specific to however you want to handle textures in your project.