I don't see this happening using a watch battery and here is why. You don't mention any particular battery but as an example we can select one at random. The V364 for example is a 1.55 volt 20 mAH battery. Very typical watch battery.
To get what you want first consider the parts needed. The first part being a sensor to detect movement, a PIR (Passive Infrared Sensor) sensor would be a good start.
This unit is an example of a PIR sensor.
This unit works great from 5 to 12V (datasheet shows 12V). You can also install a jumper wire past the 5V regulator on board to make this unit work at 3.3V. Sensor uses 1.6mA@3.3V.
The alarm pin is an open collector meaning you will need a pull up resistor on the alarm pin. The open drain setup allows multiple motion sensors to be connected on a single input pin. If any of the motion sensors go off, the input pin will be pulled low.
Best case and this is true of most of these units is a 3.3 volt supply and in a quiescent state it draws 1.6 mA (not bad) but since you want to detect a lack of motion it will always be on when there is motion likely increasing the current draw. Anyway, you need a 3.3 volt minimum. There are batteries that will do it. Home computer CMOS batteries come to mind but a little larger than a tiny watch battery.
Once you choose a sensor we need additional circuitry. Now before getting into this remember something. The more circuitry you add the more current you need. Your sensor simply tells you if there is or is not movement.
Given a choice at this point I would look to using a small simple micro-controller chip. This means if you are unfamiliar with PIC chips you will need to do some learning and program it but all your timing and everything can be done in this single package. Fewer parts, less power demand.
Now you can go with discreet components using timers and whatever but you are adding more parts. More power needed and consumed.
Overall I don't see using a watch battery as practical for what you want to do unless you use a micro-controller and learn a little basic programming.
Just My Take
Ron