Yes , you can do UX/UI design without a computer science degree. You can get the resources online or in any private institutions which provide you knowledge about the course and provide you with a certificate . The company looks for the skills and knowledge , but someone with a computer science degree will have a higher preferences for an interview in a company and also will have an advantage to get into a good company easily with more options available ,considering both have equal ui/ux design knowledge . 

For example : Consider you as an interviewer , what will you prefer , a student with a COMMERCE DEGREE WITH A UI/UX DESIGN CERTIFICATE OR A COMPUTER SCIENCE ENGINEERING  DEGREE ? Of course a student with a computer engineering degree , the computer science engineering degree student will also have a higher package then with a non computer science student .