"Malky [Mackay, sporting director] was also just in the door in terms of the structure and the transitional period that the club was in at that time," he said.
"Would I have expected [criticism] to come on quicker now because we're a bit more settled and we've had time to do things? Naturally. I'm not averse to criticism, I'm not shying away from it.
"I'm certainly not put off by it in terms of feeling sorry for myself. If anything, it's the opposite. "
Hibs have not been to a semi-final since losing to Aberdeen in the same competition in November 2023, when Gray was on the coaching staff under Nick Montgomery.
"When we spoke about challenges and objectives at the start of the season, getting to Hampden was right at the very top of the list," he added.
"If I allow the noise to affect what we're doing or distract me in any way then you just go under in that situation.
"I've been through adversity before and the focus is simply that we need to make sure we win.
"The form and everything else doesn't matter this weekend. It's about making sure everyone knows the job in hand, which is a real chance to get to Hampden. "
County are unbeaten in League One and one of veteran striker Brian Graham's nine goals came in a home draw with Dundee in the competition's group stage.
"They've stayed full-time and kept their core together," Gray said of the Highland side relegated from the Championship. "It wouldn't surprise me if they found themselves back in the Premiership again with the infrastructure, players and management team they've got.
"They've started the season really well, they're used to winning games and they'll be playing with confidence. That adds to the challenge we've got. "
