No, most spare tyres do not have a TPMS sensor installed. This is because a spare tyre is only intended as a temporary replacement. In the event of a flat or damaged tyre. They should not be driven on for long distances.
Having said that, some vehicles with a direct TPMS system may come equipped with a TPMS sensor on the spare.
If this is the case for you, the TPMS sensor on the spare tyre will only send pressure readings to the vehicle's computer if spare tyre is in use. They will not monitor the pressure of the spare tyre when it is being stored in the boot. If you're not sure whether your spare tyre has a TPMS sensor you can check your car’s handbook. Failing this, contact the manufacturer for more information.