The extra cash over the course of this parliament is a key part of the party's new policy on mental health which was launched in Edinburgh on Monday.
The policy calls for parity of esteem between mental and physical health as well as for round-the-clock mental health support in every A&E department.
Further measures outlined include a new £10 million community mental health development fund and plans to have mental health link workers in GP practices.
The Conservatives also want to help more disabled people with mental health issues into work, examine how social media companies can help tackle online bullying and appoint mental health champions in schools and businesses.
Their policy also pledges to hold a review of services for pregnant women and new mothers with a view to expanding ante-natal and post-natal mental care, and to create an improved referral system for young people who self-harm so they can be seen without delay.
Official figures published last week showed nearly eight out 10 (78.8%) children and young people seeking help with mental-health problems between July and September were seen within the Scottish Government's 18-week target, a slight improvement on 77.6% on the previous quarter but still under the government's 90% target.
A total of 100 children and young people had waited more than a year for treatment.
Scottish Conservative leader Ruth Davidson said: "For too long there's been a failure to recognise mental health issues as being just as urgent and debilitating as physical injuries.
"As we stated in our manifesto in May, there needs to be a step change in the amount of support we currently offer to those suffering from mental ill-health.
"That is why we are calling for an extra £300 million to be spent over the course of this parliament on improving services and ensuring that no-one has to wait too long to receive treatment.
The party's mental health spokesman Miles Briggs said; "More needs to be done to improve capacity and staffing across the health service and address the unacceptable waiting times for treatment.
“We now hope that SNP ministers will listen to these proposals and consider them ahead of the publication of the new mental health strategy next year.”
A spokesman for the Scottish Children's Services Coalition welcomed the new policy document.
He said: "We know that half of all diagnosable mental-health problems start before the age of 14 and 75% by the age of 21.
"As such, it is vitally important that we radically improve mental health services and increase investment in these, with an overall aim of ensuring that children and young people get the help they need, when they need it."
Scottish Liberal Democrat health spokesman Alex Cole-Hamilton said "nobody can trust" the Conservatives on mental health.
He said: "After they came to power in the UK with a majority in 2015, they abandoned many of the new mental health initiatives they had promised.
"They scrapped the scale of the promised investment in children's mental health care and they failed to implement the flagship new access standards in mental health which came into force in April."