Unfortunately, currently the saved query can only be used by the current Application Insights instance. A user feedback is already raised here.
You can consider using workbook(Note that workbook is not designed for this and has some limitations, but we can use it to save query and use the query in other Application insights instances).
Steps are as below:
1.Nav to azure portal -> one of your application insights -> click Workbooks -> create an empty template:

2.Click Add -> then click "add query":

3.In the new page, you can select one of your Application Insights from the Resource dropdown
-> then write your query code -> then click the "Run Query" button to check the results-> click Save button to save the workbook:

4.Next time, if you want to re-use the query written in step 3
, just open the saved workbook -> click the Edit button to enter into edit mode:

5.Then click the Edit button in edit mode:

6.In the new page, click the Change button to select another Application Insights. Then the query will be based the new selected Application insights:
