Welcome to the forum. Responding first to the question you posed in the title, before reading anything else: Yes, gonorrhea goes away on its own. In the pre-antibiotic era, it was generally believed that urethral gonorrhea in men cleared up without treatment in an average of 6 weeks or so. Genital infection in women took longer, but was believed to usually resolve in 3-6 months. More prolonged infection probably occurred, but not very often.
Now I have read the question itself. It is exceedingly unlikely that anyone ever carries gonorrhea for a year or more, and probably it never happens in men. Gonorrhea without symptoms (of any duration) is rare in men. As for your wife's "fertility problems", gonorrhea is an exceedingly uncommon cause of female infertility.
Chlamydia is a more likely STD than gonorrhea in this scenario, but that also is very unlikely. However, it lasts longer than gonorrhea and can be carried by women for at least a couple of years.
The tests for both gonorrhea and chlamydia are reliable in the absence of symptoms. Since you are concerned, a urine test for both. You can expect negative results.
I hope this helps. Best wishes-- HHH, MD