How return error message in spring mvc @Controller

As Sotirios Delimanolis already pointed out in the comments, there are two options:

Return ResponseEntity with error message

Change your method like this:

@RequestMapping(method = RequestMethod.GET)
public ResponseEntity getUser(@RequestHeader(value="Access-key") String accessKey,
                              @RequestHeader(value="Secret-key") String secretKey) {
    try {
        // see note 1
        return ResponseEntity
            .status(HttpStatus.CREATED)                 
            .body(this.userService.chkCredentials(accessKey, secretKey, timestamp));
    }
    catch(ChekingCredentialsFailedException e) {
        e.printStackTrace(); // see note 2
        return ResponseEntity
            .status(HttpStatus.FORBIDDEN)
            .body("Error Message");
    }
}

Note 1: You don't have to use the ResponseEntity builder but I find it helps with keeping the code readable. It also helps remembering, which data a response for a specific HTTP status code should include. For example, a response with the status code 201 should contain a link to the newly created resource in the Location header (see Status Code Definitions). This is why Spring offers the convenient build method ResponseEntity.created(URI).

Note 2: Don't use printStackTrace(), use a logger instead.

Provide an @ExceptionHandler

Remove the try-catch block from your method and let it throw the exception. Then create another method in a class annotated with @ControllerAdvice like this:

@ControllerAdvice
public class ExceptionHandlerAdvice {

    @ExceptionHandler(ChekingCredentialsFailedException.class)
    public ResponseEntity handleException(ChekingCredentialsFailedException e) {
        // log exception
        return ResponseEntity
                .status(HttpStatus.FORBIDDEN)
                .body("Error Message");
    }        
}

Note that methods which are annotated with @ExceptionHandler are allowed to have very flexible signatures. See the Javadoc for details.