Lennon could hear the shouts as he walked into his post-match media conference and said: "It doesn't make me feel good obviously. We are not in a good moment.
"The game is a small microcosm of the run we are on: penalty and set-play. We have given the opposition free hits.
"We dominated the game and lacked a bit of quality in the final third."
Ross Stewart's 39th-minute penalty and Alex Lacovitti's late header earned County a 2-0 second-round win to ensure Celtic's run of domestic success will finish this season.
It was a first cup defeat since Ronny Deila's side lost to Mark Warburton's Rangers in the semi-final of the Scottish Cup in April 2016.
Celtic can make it 12 consecutive trophies if they beat Hearts next month in last season's delayed Scottish Cup final but their 35-match winning run in domestic cups ended with an uninspiring display at Celtic Park.
When asked what would he would say to the supporters outside, Lennon said: "What can I say? That's their opinion. It's been rumbling for a while.
"It doesn't matter what I say or what sort of bravado I put on. It won't wash. I have to turn it round with results. I can only do that with the players."
When asked whether he gave himself a time frame for that, Lennon said: "No. Next game possible, whether it be AC Milan or St Johnstone next week."
When asked if he would understand if the Celtic board took the decision to remove him tonight, Lennon said: "No, I think it's too soon for that.
"We have lost one trophy, that's gone. Our priority is the league, it always is, every time we come into a season.
"We are out of Europe, so it's not been great, but there's plenty of time to turn it around. But it's alright me saying that, we have to show it with action now.
"I think I should get more time at it but, if not, then so be it."