I recently purchased a Nest Thermostat (3rd Generation). My old thermostat was working perfectly. I wired Nest the Same exact wiring as the old thermostat which is : Y1 W1 G RH RC. With Nest, the indoor fan blower randomly turns off for 3-10 seconds, then back on for few to several minutes. I replaced my indoor fan, thinking its a blower issue and still doing the same thing. I replaced my old thermostat, its working perfectly, but only when i use Nest, the Blower randomly shuts off then on..
The battery is likely depleted and it's pulling a small amount of power to charge the battery. This signal is causing the blower to turn on momentarily. The fix is probably to run a common wire, preferably to the A/C since that's where Nest expects it in a dual transformer configuration (which I assume you have since both Rh and Rc are connected).
Hopefully your A/C control board has a common terminal, and an unused wire. If not, you can use a Venstar Add-A-Wire.
Does the Nest shut off and turn back on when this happens? It's possible it is consuming too much power and not leaving enough for the relays on the control board... In my case, this happened when the "backlight" of a screen turned on, it drew too much power, and it shut off.
IIRC -- Nests turn on the screen when you walk into the room. Might be triggered by people, or if a return is too close to the nest where it senses the "heat" movement.