I see several problems with that schematic. One is that the first stage has an inductor directly across its inputs. The DC feedback around that stage will only suffice to get the output active if the input offset is small enough that, when multiplied by 3.3M/Rcoil, the result is somewhat less than 1.5V. That stage should be AC coupled like the others. The second problem is that the threshold of detection is poorly determined. Whether those two independent but nominally identical voltage dividers end up producing the same level is a matter of chance. The third is that the circuit's output offset will be excessively sensitive to the op-amp bias current. The LM324 is not the best choice given that sensitivity. A fourth problem, related to the second, is that it will detect pulses in one direction a lot better than in the other direction. But there is nothing in the setup as described that justifies any particular polarity of the signal produce by nearby pet motion. A fifth issue is not really a problem, but a possible improvement. The use of a relay coil for a pickup is not really clever and probably gives less sensitivity than could be had with a larger pickup coil made with a few wraps of wire surrounding a larger area.
The problem with polarity can be mitigated with some experimentation and arranging that the magnet placed on the pet is in a (more or less) known best orientation. The result of those experiments may also influence the coil orientation relative to where the pet is expected to be when this gizmo is activated.