Not if the objects are the same size
It has been stated elsewhere by Jeremy Crawford that illusions which hide something do not hide the effects of that thing, specifically the emitting of light.
The invisibility spell doesn't prevent you or your gear from emitting light, yet that light makes you no less invisible. The light appears to be coming from the air. Spooky!
The aura is around the object in question: by creating an illusion of the same object (but mundane) you create the image of the object but not the air around it: so the aura would remain.
However, you could certainly create an illusion of a larger object, and hide the aura within it (such as creating the illusion of a basket around a conjured knife). The DM would have to decide how large the aura is, but it presumably is not larger than a 5' cube. So as long as you're willing to create a large enough illusion (such as a 5' cube of granite) you can definitely hide your conjured object's aura within it.
Keep in mind though: the illusion is fixed in space and cannot move (for that, you'd need silent image or some higher level spell). So although you could conceal the object with this illusion, you couldn't actually do anything with the conjured object while making it appear normal. You'd probably be better off just putting the object inside your shirt.