Combat Stress will receive £1.4 million and Veterans First Point is being given £666,000.
Mr Brown said: “In addition to the pressures caused by the pandemic, in August last year the withdrawal from Afghanistan affected veterans across the UK.
“We know that veterans’ mental wellbeing services in Scotland received a significant increase in demand from both veterans and family members who are concerned about loved ones.”
He welcomed the government’s publication of a new action plan for veterans’ mental health.
Mr Brown was a fresh-faced Royal Marine commando, aged just 19, when he sailed to the Falklands in 1982 following the Argentine invasion of the islands.
He added: “The government endorses the proposal that veterans should be able to access services easily and at the right time.
“When they do, those who provide services to veterans should understand their needs.”
Speaking in the debate, Scottish Conservative MSP Craig Hoy said there should be “collective shame” over the mental health issues faced by veterans.
He said: “The Scottish Conservatives will always stand up for our armed forces, some of whom today are headed to eastern Europe to support Nato as it reinforces its eastern flank.”
Mr Hoy added: “Let us be in no doubt that the bombs and missiles which accompany the drumbeat of war in mainland Europe today will undoubtedly affect many of our own veterans.
“It will reawaken past trauma and it will open old wounds.”
Scottish Labour’s Paul Sweeney said: “As someone who has been a member of the Army Reserves for over a decade, I know on a deeply personal level the sacrifices members of our armed forces make.
“Their role is a vocation, one which means spending huge periods of time away from family and friends, and one which often leaves them isolated from civilian life.”
He called for veterans at risk of suicide to be specifically considered in the government’s upcoming suicide prevention strategy.