My mom took beetroot juice for high bp and then got gout for the first time ever. can this be due to its fructose converting to uric acid?

No: Fructose is a sugar and is not metabolized to uric acid.

Beet fructose: Yes. Kidney impairment and diuretic meds are common causes, also. Diet can aggravate. So avoid alcohol, animal fats, organ meats and shellfish. Limit other meats to 4 oz per day. Avoid high fructose corn syrup containing drinks and foods. Add egg whites, quinoa, soy as protein. Drink 3 liters per day. Tart cherry juice daily can help.