So your problem is - you do not have textures, and you do not export them. what you are using is called (not sure about blender) proceduraly generated texture.
textures you export with obj 3d file should be .jpg . png etc.
so you have 2 options
1. "bake" procedural texture to make it a .jpg . tif .etc https://www.youtube.com/watch?v=LLQFopN--LY
2. use image editing soft (photoshop gimp) to create a texture and apply it in blender (color value in material) then export. when you export you should get 3 files not 2. now you have .obj ( a 3d file with geometry) .mtl (a text file that tells any software to look for specific image texture file that u do not have and a texture file, say .jpg) and you should get a third - texture file.
ps. I see you have animation on your curtains. i do not think you can export that in .obj you could export shape keys in .fbx as far as i know.