Relays operate by flowing current through a coil. The coil generates a magnetic field, which attracts a ferromagnetic plate that then closes contacts. Without current flow, no relay can operate.
This coil has a standard resistance/impedance based off the length of wire coiled, ans is engineered depending on the size of relay, designed operational voltage, etc. This cannot be modified.

With a series of solid state op amps, one could turn a “no current/infinite resistance” signal into a relay switching effect, but it would require some other outside power source to run the relays or mosfet switching. Seems complicated. Maybe look for a “solid state relay” or “digital relay”, I’ve not ever used one though.
I know you weren’t operating a car horn directly through a relay coil either, so there must definitely be some other relay or more likely a BCM computer with a MOSFET on board.

What year/model of car? Is there any chance you can locate a oem wiring diagram to understand better how the horn system works? In older cars, the horn is a ground-switched circuit, where the horn unit is always hot +12v and then it is directly connected to ground through the steering column when the contacts on the horn pad are pressed together. No intermediate relay. On a system like this I would connect the relay effectively “in parallel” with the horn, with one side of the coil on +12v hot, and the other on the non-grounded, switched side of the horn pad. This way the horn and coil sit at the same potential and no current will flow anywhere except the horn pad ground circuit when pressed.

I cannot guarantee any newer car uses a system remotely like this though, especially through a computer so YMMV.

And realistically you could in a pinch pull coil power from the horn itself. Just connect a wire to either side of the horn so your relay coil is in parallel, one side switched and the other side always hot/ground. This may/may not be safe though as the horn circuit will likely be in a relatively large fuse and it’ll be impractical to relocate the relay or run an additional 14 feet of small gauge sense wires from the cabin to the front of the car.