How to convert dBm to V/m, A/m^2 and W/m^2
If you assume all the power shown by the spectrum analyser is efficiently transferred from space to your antenna then that's a beginning. For this to be true, the antenna has to be matched to the 50 ohm input on the spectrum analyser and, that there are minimal losses. For minimal losses to occur, the antenna needs to be length-resonant at the frequency of interest i.e. a quarter wave monopole operated with the right frequency to make it quarter-wave resonant. If it isn't properly length-resonant then it's unlikely that you can assume that it's losses are insignificant and that it can be effectively matched to 50 ohm.

So, once you have overcome this hurdle you need to estimate the effective aperture (\$A_e\$) for the antenna. Given that a monopole has a gain (\$G\$) of about 2 dB we can use this formula: -

$$A_e = G\cdot\dfrac{\lambda^2}{4\pi}$$

Or just use this calculator. The result is in square metres.

So, given that the power can be determined (within the constraints mentioned above) and we know the area, we can calculate how many watts per square metre are being received. The watts per metre can be broken into these: -

$$W/m^2 = V/m\cdot A/m$$

And, because we know that the impedance of free space is approximately 377 ohms we can say this: -

$$W/m^2 = \dfrac{V^2/m^2}{377}$$

That allows you to calculate the electric field.