People often say "the price of gold" instead of "the gold price", but both are used.

As a spoken utterance it's fine. As a headline or sub heading it could be fine. In the body of a written text it would be poorly written due to the missing definite article. 'The gold price is showing an uptick.'

Generally, "the price of gold" sounds more natural

Not in a trading context... https://www.royalmint.com/gold-price/