She said equality for women was "at the heart of the Scottish government".
More details about the competition, which is being run by the Young Scot organisation, will be released at a later date.
Ms Sturgeon has also called on women in leadership roles to follow her example and become mentors.
Speaking at a diversity event in Edinburgh, the first minister said: "My ambition for all our young people is that they will have a fair chance to succeed, and if they work hard and have the ability, there's nothing to stop them realising their dreams.
"It is not acceptable in 2017 for women to be under-represented in senior positions, including in the boardroom."
Ms Sturgeon insisted that her government offered "ambitious commitments" in support of women's equality which included;
She explained that one of her first moves as first minister was to appoint a gender-balanced cabinet.
The chief executive of the Young Scot charity, Louise Macdonald, said that supporting young people in realising their career potential and meeting their ambitions was the responsibility of every leader in Scotland.
She added: "Mentoring is a powerful way of helping young people achieve their goals and will help to create a new generation of positive role models to inspire other young women in the future.
"We're delighted to support the first minister and the Scottish government with this new mentoring programme and we are encouraging as many young women from across Scotland as possible to apply for this amazing opportunity."
"I hope that my election as first minister does indeed help to open the gate to greater opportunity for all women.
"I hope that it sends a strong, positive message to girls and young women, indeed to all women, across our land - there should be no limit to your ambition for what you can achieve.
"If you are good enough and if you work hard enough, the sky is the limit and no glass ceiling should ever stop you from achieving your dreams."
Scotland's First Minister Nicola Sturgeon has launched a competition to find a woman aged between 18 and 23 to mentor for a year.