This would not be exact, but a reasonable approximation. Assuming your structure of interest has label 4, do the following in c3d:
c3d my_seg.nii.gz -thresh 4 4 1 0 -voxel-sum -dilate 0 1x1x1 -voxel-sum
This will print the volume of the structure of interest before and after erosion by 1 voxel. Essentially, erosion takes the outer layer of voxels off, which corresponds to surface area. If your voxels are anisotropic, this won't be correct, and you would need to first resample your image (using -int 0 to get nearest neighbor interpolation and -resample-mm) to isotropic resolution.
The other approach is to export your segmentation as a mesh, and look for mesh visualization tools that can report surface area. I am sure some do, but don't know off the top of my head.