The Nest Protect shouldn't beep intermittently. It will chirp continuously once the battery is almost exhausted.
From the online support documentation as of this writing:
When the batteries are getting low:
You can also check the battery life of your Protect at any time.
Most likely it's a low battery issue. Assuming it's a battery only not hardwired protect. I had the same thing happen when 1 of the 6 batteries went bad. It skipped all the other warnings and went straight to the chirping.
You can test the batteries with a battery tester or multi-meter, but for a safety device it's usually a good idea to just buy 6 new energizer lithium AA batteries and replace them all.