The young Magpies go head-to-head with the Rams at Whitley Park following a 2-1 win against Reading in the last-16 meeting at Woking.
United's cup opponents sit in third place of Premier League 2's top division but Watts believes the underdog tag does not apply to United's youngsters - who ply their trade in Division Two - and feels the squad are more than capable of earning a semi-final spot.
Watts told nufc.co.uk: "They are a good team in a division above us so we know it will be a tough contest but I think the way we are performing game by game in the league and in other cups, we are more than ready to face the test. I think Derby will come with the right mindset and just because we're in a division below them, it doesn't mean it will be a walkover.
"We have beaten two very good teams (Reading and Wolves) and getting wins against those who are higher than you in the table is always positive. The camp certainly has a lot of confidence going into the cup game.
"We're in the knockout stages and, with cup matches, it's a one-off game. I don't think we are looking past the quarter-final and the squad are fully focused on playing Derby on Monday and we want to win the game.
"Some of our best performances have come at Whitley Park. It's a ground we have won many games at this season and it seems that opponents don't want to come here to play against us."
United reached the quarter-final stages of the Premier League Cup for the third successive season but have yet to venture beyond the last eight after defeats to Sheffield United and Reading respectively.
Newcastle took a step closer to securing the play-offs in Premier League 2 following a 2-0 win over league leaders Wolves, with forward Tyrique Bartlett breaking the deadlock after 40 seconds and Watts thinks that the opening goal could play its part in the cup tie if United are to reach the last four of the competition since its formation in 2015.
He added: "The first goal in any game is hugely important. To get two in the first seven minutes against Wolves was even better.
"I think Derby will be ready from the kick-off but I think that when we go ahead, many teams struggle to get a result against our side. We are very stable and compact when when leading, very hard to break down with plenty of playing styles.
"There has been a bit more experience in the past years with the under-23s and this is quite a young team. However, I think everyone loves playing for the team and wants to give it their best in every match. You can see the enthusiasm in training which passes into games and the mood and energy is always high."
After signing his first professional contract with the club last summer, the 19-year-old - who originally played as a midfielder for the under-18s - revealed his challenging transition in moving from central midfield into a defensive role.
Now playing as a key figure in the back line for Newcastle's youngsters, Watts has learnt a great deal in his newly adopted position.
"I've really enjoyed it," Watts said. "All the lads have been great, such as the senior members who are still here or out on loan. It was great to be playing with Sean Longstaff at the start of the season and seeing where he is now after doing so well for the first team.
"Ben (Dawson) has helped me massively as playing in centre-back has been a new position for me. I've managed to play for the majority of the season in that role so he has worked with me day-by-day after every game and in training to help me.
"It's been different. The main thing was trying to find out my strengths as a defender and using them as much as possible. I've tried to learn as quickly as I can and it has been a challenge that I think I've done well in.
"Because it is a new position, I want to keep learning everyday. I don't feel I have ever hit a wall because I feel I can take in something new."
