Do I need to submit my social security number to be designated as a beneficiary in a TRS account?

Yes, if you are being named as a beneficiary on any kind of financial product (life insurance, retirement benefits, etc.) you will need to be identified by your social security number to ensure the company properly pays the claim once it is due to you. Many people have similar names, so the SSN allows the company to properly identify the proper party for payment when a claim is made.