Don't know of a way to log it or get the queries into the db that are used other than logging it at the db end if possible.
Did some testing and the get robots is an easy one since with nothing set and just clicking Try It Out it returns all the robots. Note that each hub is also a robot, so an instance with 10 robots and 2 hubs will return 12 robots. Filters can be used, and that can get complicated, but details were not provided.
Note this from the documentation:
Encoding Request Parameters
For GET and DELETE request parameters you need to encode the 'Key' using an encoding tool before you click Try it out! For example, if the 'Key' value is /disk/alarm/fixed/C:\/warning/active, then the encoded value is %2Fdisk%2Falarm%2Ffixed%2FC%3A%5C%2Fwarning%2Factive which should be used as the 'Key 'request.
uimapi APIs - CA Unified Infrastructure Management Probes - CA Technologies Documentation
Since a login is required to use uimapi, it should be verified the client used a uim admin account.
Another option is to compre the results with what webservices_rest returns.
webservices_rest APIs - CA Unified Infrastructure Management Probes - CA Technologies Documentation
But probably the best place to start is getting exactly what the client entered, or do a webex and have him show you.