County are embroiled in a five-way fight to avoid a two-match play-off for top flight survival; Derek Adams' side are currently in 11th place - but just three points adrift of Hibernian in seventh.
Kilmarnock are the visitors to Victoria Park this weekend and Quinn is demanding total focus for the last four games of the campaign.
Quinn told the North Star: "If you don't have the mental strength for these types of games then it's tough.
"We need to stand up and be counted and dig in to get the results we need to get ourselves out of the mess we find ourselves in.
"It was a really sore result to take at Hearts on Saturday, given the effort that we put in and the chances we created. To come away with nothing left a bitter taste in our mouths. It wasn't a good day for us.
"The game was a bit scrappy throughout, but I did think we tested their keeper quite a lot and had several good periods of play.
"From the turn of the year, we have largely been playing very well and the standard in training is excellent. There's no doubting the quality, but we need to take that on to the park and get the results that we need."