I need a signal when my furnace fan turns on
As noted in the comments, depending on your furnace make/model, but assuming it's not too ancient, it likely has a signal wire coming from the furnace control board that can control a damper.

Alternatively, they do make "barometric" switches that can serve the same purpose. Installed in the return, such a switch would detect the lower pressure and activate a damper.

All that said though: opening a fresh air damper isn't necessarily going to introduce fresh air into your house, if there's not something pulling stale out out of the house. Assuming your fresh air damper feeds into the furnace return, then to the extent that the lower pressure in the return does pull in fresh air, that will reduce the draft at the return grill and cause the house air pressure to increase, pushing air out through chinks and cracks as best as can happen.

I suspect that in practice, you won't get much incoming fresh air from the damper. In any case, you might want to consider a more active approach, with a fan, or at least another exhaust damper that opens when the fresh air one does.

Bottom line: it's not really enough to just let fresh air in. You also need a solution to let the stale air out, to keep things balanced.