No, she should not start a new year for it, rather she should complete the year that she was counting for the gold that she had first.
That is because the basic principle according to the scholars is that the one who exchanges wealth that is subject to zakaah for wealth of the same type should not start counting a new year, rather he should complete the year for the first wealth.
If he exchanges gold for gold, or silver for silver, or sheep for sheep, or camels for camels, or trade goods for trade goods, he should not stop counting the year because of that, rather he should complete the first year.
Ibn Qudaamah (may Allaah have mercy on him) said in al-Mughni (4/135):
If he sells wealth that reaches the nisaab for zakaah for which he was counting the year, for wealth of the same type, such as camels for camels, cattle for cattle, sheep for sheep, gold for gold or silver for silver, he should not stop counting the year, and he should count the year for the second (wealth) on the basis of the year for the first. End quote.
Shaykh Muhammad ibn ‘Uthaymeen (may Allaah have mercy on him) was asked about a woman who had gold that reached the nisaab, and during the year she exchanged it for other gold; should she stop counting the year and calculate a new year from the time of the exchange, or not?
He replied: The year should not be interrupted in this case, because this woman exchanged gold for the same type of wealth. End quote.
Majmoo’ Fataawa Ibn ‘Uthaymeen (18/45).
And Allaah knows best.