Real Madrid beat the Hungarian side 2-1 in Budapest, with Mario Rivas heading the visitors ahead before Carlos EspĂ doubled the lead early in the second half, finishing off a combination involving Arda GĂŒler and Valverde.
Kenan Kodro pulled a goal back for FerencvĂĄros, but Mourinho's side held on for their latest pre-season win, one that also brought the second-half debut of new signing Bernardo Silva. Valverde, wearing the captain's armband, spoke to reporters in the mixed zone afterwards about the team's progress and his own evolving role at the club.
Reflecting on the performance and the project taking shape under Mourinho, Valverde struck a positive tone.
"Very good. We're continuing to polish what the coach wants, the ideas he wants. We're still missing a lot of players, but we're finding our feet better and better. We won, and that's what matters. Even though it's a friendly, the coach and the team want to win. We want to keep adding victories. The team felt good," he said.
On his early impressions of working under the Portuguese coach at Valdebebas, he added: "Good, I'm really enjoying having him here with us and trying, as I said in an interview, to learn from him, to get the most out of every day I get to train with him and his coaching staff.
"The players are happy, working really well, and I, as captain, am glad that all my teammates are doing well."
Asked what it means to now be Real Madrid's first captain, Valverde was candid about the honour and the years spent building towards it.
"Yes, obviously. In the end I've spent years being a bit of a third and second captain, and it's a role I like, that I feel a great deal of honour and pride in. I've learned a lot, I keep learning every day, and I try to give my best image both inside the dressing room and outside, on the pitch, so that all my teammates feel comfortable."
On Mourinho specifically, he was similarly effusive.
"Spectacular. Like I said, enjoying every day, trying to get the best out of myself, my best version, and knowing how to get that out of my teammates too, so that the coaching staff are happy and proud of our work. We're always going to give our maximum for him."
The midfielder also touched on Bernardo Silva's first appearance in a Real Madrid shirt, a player he will now share the centre of the pitch with.
"Spectacular, honestly, really enjoying it. He's a magical player day to day, an excellent person, who fit in really well with the whole team, very funny and entertaining. He's a top, top player who I've always admired as a player and as an opponent, and being able to have him with us today is a great joy."
Asked what has surprised him most about Mourinho since his arrival, Valverde pointed to a side of the coach that contrasts with his public reputation.
"That he's very close to us. In the end he's a person who's right there with his character, tough at times, but I think that's the way that can reach someone best. I like how he treats us, how he treats me. I'm happy and, as I'm captain, I try to reflect the best of myself so that my teammates and the coaching staff feel comfortable with us and we feel comfortable with him."
Finally, asked about the club's six signings this summer and Rodri's decision to join Barcelona rather than Real Madrid, Valverde kept his focus firmly on his own dressing room.
"I speak for myself, I'm focused on my club and my teammates. I have a lot of responsibility as captain and I have to always give my maximum for them. Every day I try to make sure all the players in the dressing room are content, happy and have good energy. So I worry more about them."
Real Madrid's pre-season continues on Wednesday against Deportivo La Coruña in the Teresa Herrera Trophy.