"mentioned their desire" is correct and usable in written English.
You can use it whenever you are referring to something someone has said that expresses their hopes or intentions. For example, "The couple mentioned their desire to move to the country eventually."
Early this month, as the McGreevey administration desperately sought to win legislative support for a plan to preserve a 400,000-acre section of northwest New Jersey known as the Highlands, lawmakers from the south mentioned their desire to develop Petty's Island, according to Micah Rasmussen, the governor's spokesman.
Many staff mentioned their desire to be appreciated, to receive appreciation letters rather than warning letters.
However, a few women, most of whom were refugees, mentioned their desire to die in their countries of birth.
The majority of participants, in particular young male and female smokers, mentioned their desire to quit and attempted to several times, before giving up and smoking again.
They mentioned their desire to stop smoking and alluded to the fact that this decision and taking action toward it was in their own hands, as illustrated by the below quote.
A minority of people who mentioned enjoyment of the course mentioned their desire to stop as a motivation for attendance, thus intrinsic motivation to attend often seems to be unrelated to their motivation to quit.
They cite their foster parent application to the agency, which mentions their desire to adopt a child and raise him or her in a Christian home.
A part of the excitement of traditional third-person narrative is the access it gives to the lives of fictional strangers; in a way, it's quite comforting to imagine that a guiding consciousness is moving you through people's homes and places of work, not to mention their fears and desires!
Smaller numbers also mentioned the desire for variety in their working life.
Respondents spoke about coming to Ireland because their spouse was working or had been offered work in Ireland, while others mentioned a desire to live close to family or friends in Ireland or the UK.
(Health center nurse, NGO) Several nurses mentioned the desire to work in the public system because it is where they can best serve their communities.
