I recently heard that eating a lot of protein in your diet can later increase your chances of getting gout. I was wondering if this is true.

Uric acid synthesis. Uric acid synthesis is accelerated with more protein in the diet. This does not neccesarily translate into a fact one would get gout simply by eating too much protein. Similiarly, asymptomatic elevated uric acid is not a disease.
Protein & gout. Protein in general is not the problem with gout; it's specific proteins (purines) that occur in organ meats (liver, kidney, sweetbreads), brewer's yeast (beer), and legumes (peanuts, peas). However, any high protein diet, without adequate fluid intake, can damage the kidneys. Your fp can test your blood & urine for markers of gout and/or kidney damage.
Gout risk. Gout risk is related to diet. Diet basics to lower risk are -- Avoid alcohol, animal fats, organ meats, anchovies, sardines, tuna, salmon, and shellfish. Limit other meats to 4 oz/day. Avoid high fructose corn syrup containing drinks and foods. Add egg whites, quinoa, soy as protein. Eat lots of citrus fruit and veg. Drink 3 liters fluid / day. Maintain ideal wt. .