What nails should I use to attach Hardie plank lap siding to 3/4" furring strips in concrete wall?
I'm a little lost. First, what are you using to attach the furring strips to the concrete? I'm  hoping some construction adhesive and a few concrete screws/anchors to bear the weight You could use the same thing to attach the Hardee-board directly to the concrete (with a layer of moisture barrier separating them to keep moisture from the concrete seeping into the siding). I'm also not sure why you're telling us what the garage door is framed with if these 2x6s aren't a substantial part of the anchor for the furring strips.

Anyway, assuming it cannot be done any other way, I'd say that using 1" nails where 1-1/4" is recommended is acceptable; maybe use a few more along each furring strip to compensate for the decreased "bite" of the shorter length of each nail. Understand that a pneumatic nailer will drive the nails down beneath the surface of the Hardee-board, so even with 1/16" clearance that 1" nail may contact the concrete. The only additional recommendation I would give is to nail in at a slight downward angle where possible, as if you were hanging a picture. The angle will not only give you a little extra depth to drive the nail, but the angle will increase the leverage the nail has to hold the siding, like using a nail to hang a picture.