@RestResource(urlMapping='/Accounts/*/contacts')
global with sharing class AccountManager {

    @HttpGet
    global static Account getAccount() {
        RestRequest request = RestContext.request;
        // grab the accountId from the end of the URL
        String accountId = request.requestURI.substringBetween('Accounts/', '/contacts');
        Account result = [SELECT ID,Name,(SELECT ID,Name FROM Contacts) FROM Account WHERE Id = :accountId ];
        system.debug(result);
        return result;
        
    }
}

@IsTest
private class AccountManagerTest {

    @isTest static void testGetAccount() {
        Id recordId = createTestRecord();
        // Set up a test request
        RestRequest request = new RestRequest();
        request.requestUri =
            'https://na1.salesforce.com/services/apexrest/Accounts/'+ recordId+'/Contacts';
        request.httpMethod = 'GET';
        RestContext.request = request;
        // Call the method to test
        Account thisAccount = AccountManager.getAccount();
        // Verify results
        System.assert(thisAccount != null);
        System.assertEquals('Test record', thisAccount.Name);
    }

   

    // Helper method
    static Id createTestRecord() {
        // Create test record
        Account accountTest = new Account(
        Name='Test record');
        insert accountTest;
       
        Contact con1 = new Contact(
        AccountId = accountTest.Id,
        firstName = 'test',
        lastName  =  'tester');
        insert con1;
        return accountTest.Id;
    }          

}