That should work, but the power supply is a bit under-rated for current or power.
The power supply has an output voltage of 18V and can supply up to 1.3A. Power is voltage multiplied by current, so 18 × 1.3 is 23.4 watts. That is the maximum power your power supply is rated to supply, according to the rated current of 1.3 amps.
As you said, the motor will only draw the current it needs. At 25W, your motor will be drawing slightly more than the power supply is rated for. This is not good; it's normal to use a power supply that is comfortably over-rated. For example, a power supply rated for 18V 2A, which is 36W. That's plenty higher than the 25W you need.
Edit: What type of power supply is it? Does it have the heavy transformer, or is it a switching supply (compact and lightweight)? These behave differently.
A transformer supply usually has no regulator in it. This causes poor output voltage regulation. That means that under light load (not much current being drawn from it), the output voltage will rise. It could go over 20V; maybe over 25. The output voltage is also affected by AC mains voltage variations, and the output voltage will not be clean - there will be "ripple" at twice the AC mains frequency. None of these issues are likely to be a big problem for a DC motor. When you overload a transformer supply, the output voltage drops and the ripple gets worse. Eventually if the transformer gets hot enough, it may open the thermal fuse and go dead. If it's a cheap one, it may melt!
A switching supply is fairly tightly regulated. If you draw too much current from it, at a certain point the current limit will kick in, and the power supply will start to "hiccup". The power supply starts up, detects an overload, shuts down, then tries to start up again and the cycle repeats indefinitely. This is audible as a ticking or buzzing sound. The output voltage jumps around. The motor will barely run in this state.