First you should run the Generator for at least 1/2 hour (or longer) under at least 1/2 load every month.. I normally do mine on the 1`st but may delay a bit due to an oil change I'm planing.
Second.. yes Generators typically have Carburetors Not fuel injectors, It can take a while for the fuel pump to re-fill the bowl after sitting for months.
Also, depending on the generator there is a procedure to follow. I use a modified one for best result but this is for an ONAN Emerald
Press and hold STOP till the light comes on solid (HOld for a few seconds past in fact) this starts the fuel pump and refills the carb by the way.
Then press start. (I will "Tap" start (just enough to engage starter) then back off, wait like 30 seconds and PRESS (and hold) start,, often this fires it up promptly.
But you do need to run it every month.