How do you know who had gonorrhea first?

No there is not.
You can not look at their penis or vagina to tell who got it first. Determining that is not possible. The best plan is to complete treatment, consider whether you each are adequately protecting yourself from infection in your relationship, and make any changes needed to safeguard your health.
The doctor probably will not be able to tell.
You are not able to tell who had it first.