Drive relay with IRFZ44N
The relay is 90 ohms at 12V so it means it requires a current of 12/90=133mA. According to the datasheet of IRF530 the minimum threshold voltage is 2V. You need to provide more than 2V, say 5V to provide the amount of the current that's needed, 130mA here. The transistor in in triode region so the current increases linearly with increasing \$V_{GS}\$.

At a \$V_{GS}\$ of 10V the \$R_{DS(on)}\$ is reported 0.16 ohms. Therefore the power consumption by the MOS itself would be only 2mW, which is negligible. Again you don't need a power MOSFET, a smaller MOSFET with lower threshold voltage such as 2N7000 will be fine. Also in this particular application, there is probably no need to use FETs. Any BJT can be used instead as long as you can provide sufficient current into the base. Or as commented use a Darlington pair with the base resistor above 120k ohms.



If you use a MOSFET for low switching frequency then the current into the gate would be negligible, and it would save power if you're on a battery-powered circuit.