It is possible to imitate the sound of a cricket chirping by whistling in a specific way. Create the sound by combining the trill of an "r" sound with a whistle. It may take plenty of practice to perfect the sound effect.
First create a trill by rolling your tongue as in the "r" of the Spanish name "Roberto." Now practice moving your tongue further forward in your mouth. This is creates a gargling sound. Take a break from the trilling to practice whistling. It may be easier to whistle after licking your lips.
The trick to creating the chirp lies in combining the trill, or gargle, with the whistle. Keep aiming to bring your tongue forward in your mouth. Bringing the tongue forward raises the pitch.
Another approach is to whistle through the teeth rather than through the lips. To whistle through the teeth, part the lips gently, and say "sh" as in the word "sheep". Now add the tongue trill, and stop and start the airflow.
One more method is to whistle on the inhalation rather than as you breathe out. Now add the trill. It takes some playing around to discover the method that best works for you.