Light dimmers are designed to drive loads that are largely resistive in nature like light bulbs. They are generally not compatible with loads that are inductive. Most, if not all, AC motors are inductive type loads.
That said, whether a dimmer switch will work safely with your fan or not depends entirely upon the type of motor on the fan.
There are some types of AC motors that can work to gain speed control via a dimmer switch type of circuit. These motors are often found in AC corded power tools such as electric drills and jig saws. Just as often these tools come with a variable speed capability via a proportional squeeze trigger. These tool speed controls work just like a common lamp dimmer. The motor type involved is generally the type with brushes.
Common low cost fans do not use brushed type motors. The usage life time would be just too short and non-brushed low power motors are often way cheaper to manufacture.
With this information in hand you may be able to evaluate your fan and decide if it is suitable for use with a dimmer. I suspect that it is likely not compatible.