No, being the contact of someone who has close contact with a person who has COVID-19 does not warrant testing. For example, you have a coworker whose family member is a confirmed case. You would not need to be tested. Despite coming into contact with the coworker, you did not have close contact with the person who actually has COVID-19. What you can do it monitor yourself for symptoms and practice good hand hygiene.
If you notice that you have symptoms and feel that you need to be medically evaluated, contact your health care provider and stay home/do not go to public gatherings, work/school.