I would fabricate a 5-sided box cover from extruded polystyrene insulation board. Assemble the pieces together with polyurethane construction adhesive in the joints that are tacked together with 10-penny nails (inserted by hand). The fan is likely framed in with 2x's, so 1-1/2 insulation board would be a likely choice to use. Size the insulation board pieces to make the final product match the 2x framing. Apply foam weatherstripping to the bottom face of the cover, place the cover over the fan assembly, and lay a piece of weighted plywood on top to maintain the seal.

Attic Fan Cover

I bought 8’x4’ R-5 one inch thick styrofoam board and made a box to cover mine. I just used a tube of construction glue. For the box top cover I was able to cut 4 covers from the board and that gave me a total R-20 insulation. I had some extra R-30 batt insulation so I cut to size and placed it on top of the box for some more weight and R-50 at that opening cover.

I also made a plywood box for my attic stairs opening to put an R-38 cover on it. More insulation and less air leaks.

Plain insulation doesn't seal air gaps, nor does wood. You need some kind of weatherstripping and sealing of the "boxes" to do that.

The problem with whole house/attic fans is that unless the opening to the attic is well sealed when not in use, it introduces a large air gap that negatively affects indoor comfort when it's either hot or cold outside or in the attic.

Also making good straight cuts and with proper glueing would give you an airtight box. You can do a light test to check for small gaps around the box and around attic opening and seal the gaps as needed.