Unit derivation concerning Nernst equation
M. Farooq have given good description of how to analyse units in a equation. Therefore, your assumption of the unit of $\dfrac{RT}{nF}$ should be Volts is correct. However, your question is not to assume but to derive. What's that means is unit conversion. Let's see the units of all four variables:

$R$ is a constant: $\pu{8.314 J mol-1 K-1}$
$F$ is also a constant: $\approx \pu{96485  C mol-1}$
$T$ is a variable: $\pu{K}$
$n$ is number of total electrons in the redox reaction involved, and hence a variable but unit less.

Now lets see the unit conversion:
$$\frac{RT}{nF} = \frac{\pu{J mol-1 K-1}\times \pu{K}}{\pu{C mol-1}} = \frac{\pu{J}}{\pu{C}} $$
By definition (Ohm's law):
$$ V = \frac{\text{potential energy}}{\text{Charge}} = \frac{\pu{J}}{\pu{C}} $$
$$\therefore \:\; \frac{RT}{nF} = \frac{\pu{J}}{\pu{C}}= \pu{V} $$