Maybe, especially using Occlumency to hide what’s truly desired.
I don't think so, but it might theoretically be possible. There's no official word on whether the Mirror of Erised could be fooled. But it may be possible. Using Occlumency to hide someone's true desires from the Mirror of Erised might have worked. It can shut down feelings and memories that show something is a lie, and this may work to hide the truth from the Mirror of Erised as well.
“The Dark Lord, for instance, almost always knows when somebody is lying to him. Only those skilled at Occlumency are able to shut down those feelings and memories that contradict the lie, and so can utter falsehoods in his presence without detection.”
- Harry Potter and the Order of the Phoenix, Chapter 24 (Occlumency)
Quirrell didn't know that the Mirror would have only worked for someone who wanted to find but not use the Philosopher's Stone, and no one else who looks into the Mirror has any reason to hide their true desire. (Dumbledore was able to keep his secret from Harry in a much simpler way - lying and saying he saw socks.)
Occlumency has fooled some very powerful wizards over the course of the seven books. Severus Snape was able to keep his true intentions secret from the Dark Lord for years. A few moments looking into a mirror would be relatively easy for someone skilled in Occlumency. This might not have helped Quirrell, because it's not known whether he could use Occlumency, but it definitely could be possible.